Add files using upload-large-folder tool

.gitattributes

@@ -730,3 +730,57 @@ JtAyT4oBgHgl3EQf5_rl/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex
 GdAzT4oBgHgl3EQfxf4V/content/2301.01737v1.pdf filter=lfs diff=lfs merge=lfs -text
 mdAyT4oBgHgl3EQf_vr-/content/2301.00916v1.pdf filter=lfs diff=lfs merge=lfs -text
 xtE2T4oBgHgl3EQfMAY2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+5NE0T4oBgHgl3EQfvgGE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+FdE4T4oBgHgl3EQffw1T/content/2301.05110v1.pdf filter=lfs diff=lfs merge=lfs -text
+WNAzT4oBgHgl3EQfKPsc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+R9AyT4oBgHgl3EQf7_qI/content/2301.00849v1.pdf filter=lfs diff=lfs merge=lfs -text
+M9E3T4oBgHgl3EQfwgt2/content/2301.04703v1.pdf filter=lfs diff=lfs merge=lfs -text
+M9E3T4oBgHgl3EQfwgt2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+G9E4T4oBgHgl3EQfHwwY/content/2301.04905v1.pdf filter=lfs diff=lfs merge=lfs -text
+4dE3T4oBgHgl3EQfowpv/content/2301.04636v1.pdf filter=lfs diff=lfs merge=lfs -text
+R9AyT4oBgHgl3EQf7_qI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+D9E1T4oBgHgl3EQfWgQs/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+MtE4T4oBgHgl3EQfKQyA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+bdE0T4oBgHgl3EQfngG8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+4dE3T4oBgHgl3EQfowpv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+xtE3T4oBgHgl3EQfPAlS/content/2301.04398v1.pdf filter=lfs diff=lfs merge=lfs -text
+G9E4T4oBgHgl3EQfHwwY/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+R9FJT4oBgHgl3EQfLCxT/content/2301.11467v1.pdf filter=lfs diff=lfs merge=lfs -text
+CdA0T4oBgHgl3EQfAf80/content/2301.01962v1.pdf filter=lfs diff=lfs merge=lfs -text
+p9FIT4oBgHgl3EQfwita/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+p9FIT4oBgHgl3EQfwita/content/2301.11352v1.pdf filter=lfs diff=lfs merge=lfs -text
+LNE4T4oBgHgl3EQf8A7c/content/2301.05345v1.pdf filter=lfs diff=lfs merge=lfs -text
+mdAyT4oBgHgl3EQf_vr-/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+gtE3T4oBgHgl3EQfIAmu/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+R9FJT4oBgHgl3EQfLCxT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+RtAyT4oBgHgl3EQfVPc7/content/2301.00139v1.pdf filter=lfs diff=lfs merge=lfs -text
+yNFQT4oBgHgl3EQfxzYK/content/2301.13406v1.pdf filter=lfs diff=lfs merge=lfs -text
+LtFRT4oBgHgl3EQf2Di0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+eNFAT4oBgHgl3EQf7B5d/content/2301.08742v1.pdf filter=lfs diff=lfs merge=lfs -text
+ztE4T4oBgHgl3EQfyg04/content/2301.05266v1.pdf filter=lfs diff=lfs merge=lfs -text
+L9FRT4oBgHgl3EQf2Ti5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+eNFAT4oBgHgl3EQf7B5d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vNFLT4oBgHgl3EQfjy8P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+LtFRT4oBgHgl3EQf2Di0/content/2301.13659v1.pdf filter=lfs diff=lfs merge=lfs -text
+b9FAT4oBgHgl3EQfXx3V/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf filter=lfs diff=lfs merge=lfs -text
+b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf filter=lfs diff=lfs merge=lfs -text
+xtE3T4oBgHgl3EQfPAlS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+sdAzT4oBgHgl3EQf6P79/content/2301.01874v1.pdf filter=lfs diff=lfs merge=lfs -text
+AtFAT4oBgHgl3EQfrx4U/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+y9E3T4oBgHgl3EQfPglf/content/2301.04403v1.pdf filter=lfs diff=lfs merge=lfs -text
+ztE4T4oBgHgl3EQfyg04/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+gtE3T4oBgHgl3EQfIAmu/content/2301.04330v1.pdf filter=lfs diff=lfs merge=lfs -text
+RtAyT4oBgHgl3EQfVPc7/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+yNFQT4oBgHgl3EQfxzYK/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+jNFPT4oBgHgl3EQf0zUa/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf filter=lfs diff=lfs merge=lfs -text
+g9E1T4oBgHgl3EQfzAXz/content/2301.03441v1.pdf filter=lfs diff=lfs merge=lfs -text
+Q9E3T4oBgHgl3EQfyQvd/content/2301.04719v1.pdf filter=lfs diff=lfs merge=lfs -text
+g9E1T4oBgHgl3EQfzAXz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+sdAzT4oBgHgl3EQf6P79/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vNFLT4oBgHgl3EQfjy8P/content/2301.12112v1.pdf filter=lfs diff=lfs merge=lfs -text
+LNE4T4oBgHgl3EQf8A7c/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf filter=lfs diff=lfs merge=lfs -text
+AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf filter=lfs diff=lfs merge=lfs -text
+sNAzT4oBgHgl3EQfBPpp/content/2301.00939v1.pdf filter=lfs diff=lfs merge=lfs -text

1tAyT4oBgHgl3EQfPfba/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c0ff7718807d52119985ebcaa98151ce0224b3b44aa211c0bca7fa89abcb4e88
+size 438943

1tE4T4oBgHgl3EQfaQwc/content/tmp_files/2301.05062v1.pdf.txt

@@ -0,0 +1,1362 @@
+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.
+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. Causal scrubbing: A method for rigorously testing interpretability
+hypotheses. Alignment Forum, Dec 2022. URL https://www.alignmentforum.org/posts/
+JvZhhzycHu2Yd57RN/causal-scrubbing-a-method-for-rigorously-testing.
+D. L. Donoho. Compressed sensing. IEEE Transactions on information theory, 52(4):1289–1306, 2006.
+15
+
+Tracr: Compiled Transformers as a Laboratory for Interpretability
+N. Elhage, N. Nanda, C. Olsson, T. Henighan, N. Joseph, B. Mann, A. Askell, Y. Bai, A. Chen,
+T. Conerly, N. DasSarma, D. Drain, D. Ganguli, Z. Hatfield-Dodds, D. Hernandez, A. Jones,
+J. Kernion, L. Lovitt, K. Ndousse, D. Amodei, T. Brown, J. Clark, J. Kaplan, S. McCandlish, and
+C. Olah. A mathematical framework for transformer circuits. Transformer Circuits Thread, 2021.
+https://transformer-circuits.pub/2021/framework/index.html.
+N. Elhage, T. Hume, C. Olsson, N. Schiefer, T. Henighan, S. Kravec, Z. Hatfield-Dodds, R. Lasenby,
+D. Drain, C. Chen, R. Grosse, S. McCandlish, J. Kaplan, D. Amodei, M. Wattenberg, and C. Olah. Toy
+models of superposition. Transformer Circuits Thread, 2022. https://transformer-circuits.
+pub/2022/toy_model/index.html.
+D. Hendrycks and K. Gimpel. Gaussian error linear units (GELUs). arXiv preprint arXiv:1606.08415,
+2016.
+T. Hennigan, T. Cai, T. Norman, and I. Babuschkin. Haiku: Sonnet for JAX, 2020. URL http:
+//github.com/deepmind/dm-haiku.
+K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approxima-
+tors. Neural networks, 2(5):359–366, 1989.
+A. Jacovi and Y. Goldberg. Towards faithfully interpretable nlp systems: How should we define and
+evaluate faithfulness? arXiv preprint arXiv:2004.03685, 2020.
+M. L. Leavitt and A. Morcos.
+Towards falsifiable interpretability research.
+arXiv preprint
+arXiv:2010.12016, 2020.
+K. Meng, D. Bau, A. J. Andonian, and Y. Belinkov. Locating and editing factual associations in GPT.
+In Advances in Neural Information Processing Systems, 2022.
+W. Merrill, A. Sabharwal, and N. A. Smith. Saturated transformers are constant-depth threshold
+circuits. Transactions of the Association for Computational Linguistics, 10:843–856, 2022.
+N. Nanda and T. Lieberum. A mechanistic interpretability analysis of grokking. Alignment Fo-
+rum, Aug 2022.
+URL https://www.alignmentforum.org/posts/N6WM6hs7RQMKDhYjB/
+a-mechanistic-interpretability-analysis-of-grokking.
+C. Olah, N. Cammarata, L. Schubert, G. Goh, M. Petrov, and S. Carter. Zoom in: An introduction to
+circuits. Distill, 2020. doi: 10.23915/distill.00024.001. https://distill.pub/2020/circuits/zoom-in.
+C. Olsson, N. Elhage, N. Nanda, N. Joseph, N. DasSarma, T. Henighan, B. Mann, A. Askell, Y. Bai,
+A. Chen, T. Conerly, D. Drain, D. Ganguli, Z. Hatfield-Dodds, D. Hernandez, S. Johnston, A. Jones,
+J. Kernion, L. Lovitt, K. Ndousse, D. Amodei, T. Brown, J. Clark, J. Kaplan, S. McCandlish, and C. Olah.
+In-context learning and induction heads. Transformer Circuits Thread, 2022. https://transformer-
+circuits.pub/2022/in-context-learning-and-induction-heads/index.html.
+A. Radford, K. Narasimhan, T. Salimans, I. Sutskever, et al.
+Improving language under-
+standing by generative pre-training.
+OpenAI, 2018.
+URL https://openai.com/blog/
+language-unsupervised/.
+A. Scherlis, K. Sachan, A. S. Jermyn, J. Benton, and B. Shlegeris. Polysemanticity and capacity in
+neural networks. arXiv preprint arXiv:2210.01892, 2022.
+C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich.
+Going deeper with convolutions. In Conference on Computer Vision and Pattern Recognition (CVPR),
+2015.
+16
+
+Tracr: Compiled Transformers as a Laboratory for Interpretability
+A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin.
+Attention is all you need. In Advances in Neural Information Processing Systems, 2017.
+K. Wang, A. Variengien, A. Conmy, B. 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
+

3dE3T4oBgHgl3EQfPwmd/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7fb4aa8e3302163c8a886513f7d4c76cd6d2a5d4611c8ef4e4c68fbd6dc720aa
+size 96620

4dE3T4oBgHgl3EQfowpv/content/2301.04636v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2de88b207ee1acf08548ba93af4e8eccb291ea8cead41f741e7b131ff60d8eb7
+size 209388

4dE3T4oBgHgl3EQfowpv/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:06db8f16185b32d225106f18226f7003604834cec206047e97e374253f432625
+size 2490413

4dE3T4oBgHgl3EQfowpv/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4b279c584b8e165cba4a3e3d49c2342415417b7097bdbdfa2faebcfa145391ce
+size 97257

59FKT4oBgHgl3EQfTC3B/content/tmp_files/2301.11778v1.pdf.txt

@@ -0,0 +1,809 @@
+Reproducibility of health 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: [email protected] or [email protected]. 
+ 
+ 
+ 
+ 
+ 
+
+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. The study was conceived based on previous 
+work undertaken by CG Stat for the National Association of Scholars (nas.org), New York, NY. 
+ 
+References 
+ 
+Alderson, P. (2004). Absence of evidence is not evidence of absence. British Medical Journal, 
+328(7438), 476. https://doi.org/10.1136/bmj.328.7438.476 
+ 
+Altman, D. G., & Bland, J. M. (1995). Absence of evidence is not evidence of absence. British 
+Medical Journal, 311(7003), 485. https://doi.org/10.1136/bmj.311.7003.485 
+ 
+Altman, D. G., & Bland, J. M. (2011a). How to obtain a confidence interval from a P value. 
+British Medical Journal, 343, d2090. https://doi.org/10.1136/bmj.d2090 
+ 
+Altman, D. G., & Bland, J. M. (2011b). How to obtain the P value from a confidence interval. 
+British Medical Journal, 343, d2304. https://doi.org/10.1136/bmj.d2304 
+ 
+American Institute of Economic Research (AIER). (2020). Cost of Lockdowns: A Preliminary 
+Report. AIER, Great Barrington, MA. https://www.aier.org/article/cost-of-us-lockdowns-a-
+preliminary-report/ 
+ 
+Boos, D. D., & Stefanski, L. A. (2011). P-value precision and reproducibility. The American 
+Statistician, 65(4), 213−221. https://doi.org/10.1198/tas.2011.10129 
+ 
+Boos, D.D., & Stefanski, L.A. (2013). Essential Statistical Inference: Theory and Methods. New 
+York, NY: Springer. 
+ 
+Bramstedt, K. A. (2020). The carnage of substandard research during the COVID‐19 pandemic: 
+A call for quality. Journal of Medical Ethics, 46, 803–807. http://doi.org/10.1136/medethics-
+2020-106494. 
+ 
+Centers for Disease Control and Prevention (CDC). (2007). Interim pre-pandemic planning 
+guidance: community strategy for pandemic influenza mitigation in the United States: Early, 
+targeted, layered use of nonpharmaceutical interventions. Atlanta, GA: US CDC. 
+https://stacks.cdc.gov/view/cdc/11425 
+ 
+Centers for Disease Control and Prevention (CDC). (2017). Community mitigation guidelines to 
+prevent pandemic influenza – United States. Atlanta, GA: US CDC. MMWR Recommendations 
+and reports, 66(No. RR-1)1−36. https://stacks.cdc.gov/view/cdc/45220 
+ 
+Chu, C., Baxamusa, S., & Witherel, C. (2021). Impact of COVID-19 on materials science 
+research innovation and related pandemic response. MRS Bulletin, 46, 807–812. 
+https://doi.org/10.1557/s43577-021-00186-1 
+ 
+
+Egger, M., Davey Smith, G., & Altman, D. G. (2001). Problems and limitations in conducting 
+systematic reviews. In: Egger, M., Davey Smith, G., & Altman, D. G. (eds.) Systematic Reviews 
+in Health Care: Meta−analysis in Context, 2nd ed. London: BMJ Books. 
+ 
+Gostin, L. O., Friedman, E. A., & Wetter, S. A. 2020. Responding to covid-19: How to navigate 
+a public health emergency legally and ethically. Hastings Center Report, 50, 8−12. 
+http://doi.org/10.1002/hast.1090 
+ 
+Gustot, T. (2020). Quality and reproducibility during the COVID‐19 pandemic. JHEP Reports, 
+2, 1−3. https://doi.org/10.1016/j.jhepr.2020.100141 
+ 
+Herby, J., Jonung, L., & Hanke, S.H. (2022). A literature review and meta-analysis of the effects 
+of lockdowns on COVID-19 mortality. SAE./No. 210/May 2022. John Hopkins Institute for 
+Applied Economics, Global Health, and the Study of Business Enterprise 
+https://sites.krieger.jhu.edu/iae/files/2022/06/A-Systematic-Review-and-Meta-Analysis-of-the-
+Effects-of-Lockdowns-of-COVID-19-Mortality-II.pdf 
+ 
+Hung, H. M. J., O’Neill, R. T., Bauer, P., et al. (1997). The behavior of the p-value when the 
+alternative hypothesis is true. Biometrics, 53, 11–22. https://doi.org/10.2307/2533093 
+ 
+Inglesby, T. V., Nuzzo, J. B. O’Tool, T., Henderson, D. A. 2006. Disease Mitigation Measures in 
+the Control of Pandemic Influenza. Biosecurity and Bioterrorism: Biodefense Strategy, Practice, 
+and Science. 4, 366-375. https://doi.org/10.1089/bsp.2006.4.366 
+ 
+Ioannidis, J. P. A. (2005). Why most published research findings are false. PLoS Medicine, 2(8), 
+e124. https://doi.org/10.1371/journal.pmed.0020124 
+ 
+Ioannidis, J. P. A., Tarone, R. E., & McLaughlin, J. K. (2011). The false-positive to false-
+negative ratio in epidemiologic studies. Epidemiology, 22, 450–456. 
+http://doi.org/10.1097/EDE.0b013e31821b506e 
+ 
+Ioannidis, J. P. A. (2022). Correction: Why most published research findings are false. PLoS 
+Medicine, 19(8), e1004085. https://doi.org/10.1371/journal.pmed.1004085 
+ 
+Ioannidis, J. P. A., Bendavid, E., Salholz-Hillel, M., et al. 2022. Massive covidization of research 
+citations and the citation elite. Proceedings of the National Academy of Sciences, 119, 28 
+e2204074119. https://doi.org/10.1073/pnas.2204074119 
+ 
+Ioannidis, J. P. A., Greenland S., Hlatky M. A., Khoury M. J., Macleod M. R., Moher D., Schulz 
+K. F., and Tibshirani R. (2014). “Increasing Value and Reducing Waste in Research Design, 
+Conduct, and Analysis.” Lancet 383, 166–75. 10.1016/s0140-6736(13)62227-8 
+ 
+Iqbal, S. A., Wallach, J. D., Khoury, M. J., et al. (2016). Reproducible research practices and 
+transparency across the biomedical literature. PLoS Biology, 4(1), e1002333. 
+http://doi.org/10.1371/journal.pbio.1002333 
+ 
+
+James, J. J. (2020). Lockdown or lockup. Disaster Medicine and Public Health Preparedness, 
+14(6), e6−e7. https://doi.org/10.1017/dmp.2020.127 
+ 
+Jenson, H. B. (2020). How did “flatten the curve” become “flatten the economy?” A perspective 
+from the United States of America. Asian Journal of Psychiatry, 51, 102165. 
+http://doi.org/10.1016/j.ajp.2020.102165 
+ 
+Keown, S. (2012). Biases Rife in Research, Ioannidis Says. NIH Record, VXIV(10). Nih-
+record.nih.gov/sites/recordNIH/files/pdf/2012/NIH-Record-2012-05-11.pdf (accessed on 10 July 
+2020). 
+ 
+Kosnik, L. R., & Bellas, A. (2020). Drivers of COVID-19 stay at home orders: Epidemiologic, 
+economic, or political concerns? Economics of Disasters and Climate Change, 4(3), 503–514. 
+https://doi.org/10.1007/s41885-020-00073-0 
+ 
+Kindzierski, W., Young, S., Meyer, T., et al. (2021). Evaluation of a meta-analysis of ambient air 
+quality as a risk factor for asthma exacerbation. Journal of Respiration, 1(3), 173−196. 
+https://doi.org/10.3390/jor1030017 
+ 
+Kinsella, C. M., Santos, P. D., Postigo-Hidalgo, I., et al. (2020). Preparedness needs research: 
+How fundamental science and international collaboration accelerated the response to COVID-19. 
+PLoS Pathogens, 16(10): e1008902. https://doi.org/10.1371/journal.ppat.1008902 
+ 
+Landis, S. C., Amara, S. G., Asadullah, K., et al. (2012). A call for transparent reporting to 
+optimize the predictive value of preclinical research. Nature, 490(7419), 187–191. 
+https://doi.org/10.1038/nature11556 
+ 
+Lavezzo, E., Franchin, E., Ciavarella, C., et al. (2020). Suppression of a SARS-CoV-2 outbreak 
+in the Italian municipality of Vo. Nature, 584, 425–429. https://doi.org/10.1038/s41586-020-
+2488-1 
+ 
+Magness P. 2021. The Failures of Pandemic Central Planning. October 1, 2021. 
+http://doi.org/10.2139/ssrn.3934452 
+ 
+Matrajt, L., & Leung, T. (2020). Evaluating the effectiveness of social distancing interventions to 
+delay or flatten the epidemic curve of Coronavirus disease. Emerging Infectious Diseases, 26(8), 
+1740−1748. https://doi.org/10.3201/eid2608.201093 
+ 
+Members, W.-C. J. M. (2020). Report of the WHO-China Joint Mission on Coronavirus Disease 
+2019 (COVID-19). World Health Organization (WHO). https://www.who.int/docs/default-
+source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf 
+ 
+Moonesinghe, R., Khoury, M. J., & Janssens, A. C. J. W. (2007). Most published research 
+findings are false⸺But a little replication goes a long way. PLoS Medicine, 4, e28. 
+https://doi.org/10.1146/10.1371/journal.pmed.0040028 
+ 
+
+Munafo, M. R., Nosek, B. A., Bishop, D. V. M., et al. (2017). A manifesto for reproducible 
+science. Nature Human Behaviour, 1:0021. https://doi.org/10.1038/s41562-016-0021 
+ 
+Murad, M. H., Asi, N., Alsawas, M., et al. (2016). New evidence pyramid. BMJ Evidence-Based 
+Medicine, 21(4), 125−127. http://dx.doi.org/10.1136/ebmed-2016-110401 
+ 
+Nissen, S. B., Magidson, T., Gross, K., et al. (2016). Publication bias and the canonization of 
+false facts. eLife, 5, e21451. https://doi. org/10.7554/elife.21451 
+ 
+Paez, A. (2021). Reproducibility of research during COVID-19: Examining the case of 
+population density and the basic reproductive rate from the perspective of spatial analysis. 
+Geographical Analysis, 54, 860–880. https://doi.org/10.1111/gean.12307 
+ 
+Peng, R. D., & Hicks, S. C. (2021). Reproducible research: A retrospective. Annual Review of 
+Public Health, 42, 79−93. https://doi.org/10.1146/annurev-publhealth-012420-105110 
+ 
+Piquero, A.R., Jennings, W.G., Jemison, E., et al. (2021). Domestic violence during the COVID-
+19 pandemic - Evidence from a systematic review and meta-analysis. Journal of Criminal 
+Justice, 74, 101806. https://doi.org/10.1016/j.jcrimjus.2021.101806 
+ 
+Prati, G., & Mancini, A. D. (2021). The psychological impact of COVID-19 pandemic 
+lockdowns: a review and meta-analysis of longitudinal studies and natural experiments. 
+Psychological Medicine, 51, 201–211. https://doi.org/10.1017/S0033291721000015 
+ 
+Randall, D., & Welser, C. (2018). The Irreproducibility Crisis of Modern Science: Causes, 
+Consequences, and the Road to Reform. New York, NY: National Association of Scholars. 
+Nas.org/reports/the-irreproducibility-crisis-of-modern-science 
+ 
+Schweder, T., & Spjøtvoll, E. (1982). Plots of p-values to evaluate many tests simultaneously. 
+Biometrika, 69, 493−502. https://doi.org/10.1093/biomet/69.3.493 
+ 
+Sedgwick, P. (2014). Understanding why “absence of evidence is not evidence of absence”. 
+British Medical Journal, 349, g4751. https://doi.org/10.1136/bmj.g4751 
+ 
+Seifert, J., Meissner, C., Birkenstock, A., et al. (2021). Peripandemic psychiatric emergencies: 
+impact of the COVID-19 pandemic on patients according to diagnostic subgroup. European 
+Archives of Psychiatry and Clinical Neuroscience, 271(2), 259–270. 
+https://doi.org/10.1007/s00406-020-01228-6 
+ 
+Stodden, V., Seiler, J., & Ma, Z. K. (2018). An empirical analysis of journal policy effectiveness 
+for computational reproducibility. Proceedings of the National Academy of Sciences of the 
+United States of America, 115, 2584–2589. https://doi.org/10.1073/pnas.1708290115 
+ 
+Sumner, J., Haynes L., Nathan S., et al. (2020). Reproducibility and reporting practices in 
+COVID‐19 preprint manuscripts. medRxiv, 2020.03.24.20042796. 
+https://doi.org/10.1101/2020.03.24.20042796 
+
+ 
+Ware, J. J., & Munafo, M. R. (2015). Significance chasing in research practice: Causes, 
+consequences and possible solutions. Addiction, 110(1), 4–8. https://doi.org/10.1111/add.12673 
+ 
+Young, S.S., Cheng, K.-C., Chen, J. H., et al. (2022). Reliability of a meta-analysis of air 
+quality−asthma cohort studies. International Journal of Statistics and Probability, 11(2), 61−76. 
+https://doi.org/10.5539/ijspv11n2p61 
+ 
+Young, S. S., & Kindzierski, W. B. (2019). Evaluation of a meta-analysis of air quality and heart 
+attacks, a case study. Critical Reviews in Toxicology, 49(1), 85–94. 
+https://doi.org/10.1080/10408444.2019.1576587 
+ 
+Young, S. S., & Kindzierski, W. B. (2022a). Statistical reliability of a diet-disease association 
+meta-analysis. International Journal of Statistics and Probability, 11(3), 40–50. 
+https://doi.org/10.5539/ijsp.v11n3p40 
+ 
+Young, S. S., & Kindzierski, W. B. (2022b). Research Plan Lockdowns. Researchers.One. 
+https://researchers.one/articles/22.11.00005v1 
+ 
+Zhu, Y., Li, Y., & Xu, X. 2022. 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

5NE0T4oBgHgl3EQfvgGE/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:39703860509f73df2e123c7ab4d6270c9d923c6a95b222321daf374376de510b
+size 2031661

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: [email protected]
+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. Obviously; it is fascinating to have a further study considering
+a more realistic model.
+Acknowledgments
+The work is partly supported by NSFC Grant No. 12205013. MG is also endorsed by ”the
+Fundamental Research Funds for the Central Universities” with Grant No. 2021NTST13.
+References
+[1] S. E. Gralla, D. E. Holz, and R. M. Wald, “Black Hole Shadows, Photon Rings, and Lensing
+Rings,” Phys. Rev. D 100 no. 2, (2019) 024018, arXiv:1906.00873 [astro-ph.HE].
+[2] J. Peng, M. Guo, and X.-H. Feng, “Influence of quantum correction on black hole shadows,
+photon rings, and lensing rings,” Chin. Phys. C 45 no. 8, (2021) 085103, arXiv:2008.00657
+[gr-qc].
+[3] P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, “Fundamental photon orbits: black hole
+13
+
+shadows and spacetime instabilities,” Phys. Rev. D 96 no. 2, (2017) 024039,
+arXiv:1705.05461 [gr-qc].
+[4] P.-C. Li, T.-C. Lee, M. Guo, and B. Chen, “Correspondence of eikonal quasinormal modes
+and unstable fundamental photon orbits for a Kerr-Newman black hole,” Phys. Rev. D 104
+no. 8, (2021) 084044, arXiv:2105.14268 [gr-qc].
+[5] M. Guo and S. Gao, “Universal Properties of Light Rings for Stationary Axisymmetric
+Spacetimes,” Phys. Rev. D 103 no. 10, (2021) 104031, arXiv:2011.02211 [gr-qc].
+[6] Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A* Event
+Horizon Telescope Results. VI. Testing the Black Hole Metric,” Astrophys. J. Lett. 930
+no. 2, (2022) L17.
+[7] Event Horizon Telescope Collaboration, S. Issaoun et al., “Resolving the Inner Parsec of
+the Blazar J1924–2914 with the Event Horizon Telescope,” Astrophys. J. 934 (2022) 145,
+arXiv:2208.01662 [astro-ph.HE].
+[8] Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon
+Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophys. J. Lett.
+875 no. 1, (2019) L6, arXiv:1906.11243 [astro-ph.GA].
+[9] Event Horizon Telescope Collaboration, A. E. Broderick et al., “Characterizing and
+Mitigating Intraday Variability: Reconstructing Source Structure in Accreting Black Holes
+with mm-VLBI,” Astrophys. J. Lett. 930 no. 2, (2022) L21.
+[10] Event Horizon Telescope Collaboration, M. Wielgus et al., “Millimeter Light Curves of
+Sagittarius A* Observed during the 2017 Event Horizon Telescope Campaign,” Astrophys. J.
+Lett. 930 no. 2, (2022) L19, arXiv:2207.06829 [astro-ph.HE].
+[11] Event Horizon Telescope Collaboration, B. Georgiev et al., “A Universal Power-law
+Prescription for Variability from Synthetic Images of Black Hole Accretion Flows,”
+Astrophys. J. Lett. 930 no. 2, (2022) L20.
+[12] Event Horizon Telescope Collaboration, J. Farah et al., “Selective Dynamical Imaging of
+Interferometric Data,” Astrophys. J. Lett. 930 no. 2, (2022) L18.
+[13] Y. Hou, M. Guo, and B. Chen, “Revisiting the shadow of braneworld black holes,” Phys.
+Rev. D 104 no. 2, (2021) 024001, arXiv:2103.04369 [gr-qc].
+14
+
+[14] M. Guo and P.-C. Li, “Innermost stable circular orbit and shadow of the 4D
+Einstein–Gauss–Bonnet black hole,” Eur. Phys. J. C 80 no. 6, (2020) 588,
+arXiv:2003.02523 [gr-qc].
+[15] X.-R. Zhu, Y.-X. Chen, P.-H. Mou, and K.-J. He, “The shadow and observation appearance
+of black hole surrounded by the dust field in Rastall theory,” Chin. Phys. B 32 no. 1, (2023)
+010401.
+[16] S. Chen, J. Jing, W.-L. Qian, and B. Wang, “Black hole images: A Review,”
+arXiv:2301.00113 [astro-ph.HE].
+[17] F. Atamurotov, M. Jamil, and K. Jusufi, “Quantum effects on the black hole shadow and
+deflection angle in presence of plasma,” arXiv:2212.12949 [gr-qc].
+[18] S. Sau and J. W. Moffat, “Shadow of regular black hole in scalar-tensor-vector gravity
+theory,” arXiv:2211.15040 [gr-qc].
+[19] S. Vagnozzi and L. Visinelli, “Hunting for extra dimensions in the shadow of M87*,” Phys.
+Rev. D 100 no. 2, (2019) 024020, arXiv:1905.12421 [gr-qc].
+[20] A. Grenzebach, V. Perlick, and C. L¨ammerzahl, “Photon Regions and Shadows of
+Kerr-Newman-NUT Black Holes with a Cosmological Constant,” Phys. Rev. D 89 no. 12,
+(2014) 124004, arXiv:1403.5234 [gr-qc].
+[21] S.-W. Wei and Y.-X. Liu, “Observing the shadow of Einstein-Maxwell-Dilaton-Axion black
+hole,” JCAP 11 (2013) 063, arXiv:1311.4251 [gr-qc].
+[22] V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, “Black hole shadow in an expanding
+universe with a cosmological constant,” Phys. Rev. D 97 no. 10, (2018) 104062,
+arXiv:1804.04898 [gr-qc].
+[23] X.-X. Zeng, H.-Q. Zhang, and H. Zhang, “Shadows and photon spheres with spherical
+accretions in the four-dimensional Gauss–Bonnet black hole,” Eur. Phys. J. C 80 no. 9,
+(2020) 872, arXiv:2004.12074 [gr-qc].
+[24] P.-C. Li, M. Guo, and B. Chen, “Shadow of a Spinning Black Hole in an Expanding
+Universe,” Phys. Rev. D 101 no. 8, (2020) 084041, arXiv:2001.04231 [gr-qc].
+[25] M. Wang, S. Chen, and J. Jing, “Chaotic shadow of a non-Kerr rotating compact object with
+quadrupole mass moment,” Phys. Rev. D 98 no. 10, (2018) 104040, arXiv:1801.02118
+[gr-qc].
+15
+
+[26] M. Guo, N. A. Obers, and H. Yan, “Observational signatures of near-extremal Kerr-like
+black holes in a modified gravity theory at the Event Horizon Telescope,” Phys. Rev. D 98
+no. 8, (2018) 084063, arXiv:1806.05249 [gr-qc].
+[27] J. W. Moffat and V. T. Toth, “Masses and shadows of the black holes Sagittarius A* and
+M87* in modified gravity,” Phys. Rev. D 101 no. 2, (2020) 024014, arXiv:1904.04142
+[gr-qc].
+[28] Y. Huang, S. Chen, and J. Jing, “Double shadow of a regular phantom black hole as photons
+couple to the Weyl tensor,” Eur. Phys. J. C 76 no. 11, (2016) 594, arXiv:1606.04634
+[gr-qc].
+[29] Z. Hu, Y. Hou, H. Yan, M. Guo, and B. Chen, “Polarized images of synchrotron radiations in
+curved spacetime,” Eur. Phys. J. C 82 no. 12, (2022) 1166, arXiv:2203.02908 [gr-qc].
+[30] Y. Hou, P. Liu, M. Guo, H. Yan, and B. Chen, “Multi-level images around Kerr–Newman
+black holes,” Class. Quant. Grav. 39 no. 19, (2022) 194001, arXiv:2203.02755 [gr-qc].
+[31] S. Wen, W. Hong, and J. Tao, “Observational Appearances of Magnetically Charged Black
+Holes in Born-Infeld Electrodynamics,” arXiv:2212.03021 [gr-qc].
+[32] I. Sengo, P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, “Kerr black holes with
+synchronised Proca hair: lensing, shadows and EHT constraints,” arXiv:2209.06237
+[gr-qc].
+[33] Y. Chen, G. Guo, P. Wang, H. Wu, and H. Yang, “Appearance of an infalling star in black
+holes with multiple photon spheres,” Sci. China Phys. Mech. Astron. 65 no. 12, (2022)
+120412, arXiv:2206.13705 [gr-qc].
+[34] K.-J. He, S. Guo, S.-C. Tan, and G.-P. Li, “Shadow images and observed luminosity of the
+Bardeen black hole surrounded by different accretions *,” Chin. Phys. C 46 no. 8, (2022)
+085106, arXiv:2103.13664 [hep-th].
+[35] M. Zhang and J. Jiang, “Shadows of accelerating black holes,” Phys. Rev. D 103 no. 2,
+(2021) 025005, arXiv:2010.12194 [gr-qc].
+[36] H. C. D. L. Junior, J.-Z. Yang, L. C. B. Crispino, P. V. P. Cunha, and C. A. R. Herdeiro,
+“Einstein-Maxwell-dilaton neutral black holes in strong magnetic fields: Topological charge,
+shadows, and lensing,” Phys. Rev. D 105 no. 6, (2022) 064070, arXiv:2112.10802 [gr-qc].
+16
+
+[37] Z.-Y. Tang, X.-M. Kuang, B. Wang, and W.-L. Qian, “Photon region and shadow of a
+rotating 5D black string,” arXiv:2211.08137 [gr-qc].
+[38] Y. Meng, X.-M. Kuang, and Z.-Y. Tang, “Photon regions, shadow observables, and
+constraints from M87* of a charged rotating black hole,” Phys. Rev. D 106 no. 6, (2022)
+064006, arXiv:2204.00897 [gr-qc].
+[39] G.-P. Li and K.-J. He, “Observational appearances of a f(R) global monopole black hole
+illuminated by various accretions,” Eur. Phys. J. C 81 no. 11, (2021) 1018.
+[40] S. Guo, Y. Han, and G.-P. Li, “Joule–Thomson expansion of a specific black hole in f(R)
+gravity coupled with Yang–Mills field,” Class. Quant. Grav. 37 no. 8, (2020) 085016.
+[41] D. Wu, “Hunting for extra dimensions in the shadow of Sagittarius A*,” arXiv:2205.07207
+[gr-qc].
+[42] S. Vagnozzi et al., “Horizon-scale tests of gravity theories and fundamental physics from the
+Event Horizon Telescope image of Sagittarius A∗,” arXiv:2205.07787 [gr-qc].
+[43] Q. Gan, P. Wang, H. Wu, and H. Yang, “Photon ring and observational appearance of a
+hairy black hole,” Phys. Rev. D 104 no. 4, (2021) 044049, arXiv:2105.11770 [gr-qc].
+[44] X. Wang, P.-C. Li, C.-Y. Zhang, and M. Guo, “Novel shadows from the asymmetric
+thin-shell wormhole,” Phys. Lett. B 811 (2020) 135930, arXiv:2007.03327 [gr-qc].
+[45] H. Olivares, Z. Younsi, C. M. Fromm, M. De Laurentis, O. Porth, Y. Mizuno, H. Falcke,
+M. Kramer, and L. Rezzolla, “How to tell an accreting boson star from a black hole,” Mon.
+Not. Roy. Astron. Soc. 497 no. 1, (2020) 521–535, arXiv:1809.08682 [gr-qc].
+[46] B. Kleihaus, J. Kunz, and M. List, “Rotating boson stars and Q-balls,” Phys. Rev. D 72
+(2005) 064002, arXiv:gr-qc/0505143.
+[47] B. Kleihaus, J. Kunz, M. List, and I. Schaffer, “Rotating Boson Stars and Q-Balls. II.
+Negative Parity and Ergoregions,” Phys. Rev. D 77 (2008) 064025, arXiv:0712.3742
+[gr-qc].
+[48] C. Herdeiro and E. Radu, “Construction and physical properties of Kerr black holes with
+scalar hair,” Class. Quant. Grav. 32 no. 14, (2015) 144001, arXiv:1501.04319 [gr-qc].
+[49] N. Siemonsen and W. E. East, “Stability of rotating scalar boson stars with nonlinear
+interactions,” Phys. Rev. D 103 no. 4, (2021) 044022, arXiv:2011.08247 [gr-qc].
+17
+
+[50] C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha, and N. Sanchis-Gual, “The
+imitation game: Proca stars that can mimic the Schwarzschild shadow,” JCAP 04 (2021)
+051, arXiv:2102.01703 [gr-qc].
+[51] F. H. Vincent, Z. Meliani, P. Grandclement, E. Gourgoulhon, and O. Straub, “Imaging a
+boson star at the Galactic center,” Class. Quant. Grav. 33 no. 10, (2016) 105015,
+arXiv:1510.04170 [gr-qc].
+[52] J. Baines, T. Berry, A. Simpson, and M. Visser, “Painlev´e–Gullstrand form of the
+Lense–Thirring Spacetime,” Universe 7 no. 4, (2021) 105, arXiv:2006.14258 [gr-qc].
+[53] B. Mashhoon, F. Hehl, and D. Theiss, “On the influence of the proper rotation of central
+bodies on the motions of planets and moons according to einstein’s theory of gravitation,”
+General Relativity and Gravitation 16 no. 8, (1984) 727–741.
+[54] J. Baines, T. Berry, A. Simpson, and M. Visser, “Killing Tensor and Carter Constant for
+Painlev´e–Gullstrand Form of Lense–Thirring Spacetime,” Universe 7 no. 12, (2021) 473,
+arXiv:2110.01814 [gr-qc].
+[55] J. Baines, T. Berry, A. Simpson, and M. Visser, “Geodesics for the Painlev´e–Gullstrand Form
+of Lense–Thirring Spacetime,” Universe 8 no. 2, (2022) 115, arXiv:2112.05228 [gr-qc].
+[56] Z. Hu, Z. Zhong, P.-C. Li, M. Guo, and B. Chen, “QED effect on a black hole shadow,”
+Phys. Rev. D 103 no. 4, (2021) 044057, arXiv:2012.07022 [gr-qc].
+[57] Z. Zhong, Z. Hu, H. Yan, M. Guo, and B. Chen, “QED effects on Kerr black hole shadows
+immersed in uniform magnetic fields,” Phys. Rev. D 104 no. 10, (2021) 104028,
+arXiv:2108.06140 [gr-qc].
+[58] Z. Zhang, H. Yan, M. Guo, and B. Chen, “Shadows of Kerr black holes with a
+Gaussian-distributed plasma in the polar direction,” arXiv:2206.04430 [gr-qc].
+18
+

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, [email protected]
+†Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria, [email protected]
+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
+y��
+� 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. In order to obtain estimates for the x-derivatives, for a given multi-
+index ζ, we differentiate the equation with respect to ∂ζ
+x. As the weight yα remains unchanged,
+we see that W := ∂ζ
+xU solves the extension problem (2.3)
+− 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
+�
+.
+One can modify the arguments of Proposition 2.3 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+) .
+This gives
+∥∇W∥L2(yα,Rd×R+) ≲ ∥Fζ∥L2(y−α,Rd×R+) + ∥fζ∥L2(Ω) .
+Now, an induction argument can be set up as in the proof of Proposition 2.6 to control
+∥Fζ∥L2(y−α,Rd×R+) by L2-norms of derivatives of f.
+References
+[AB17]
+G. Acosta and J. P. Borthagaray.
+A fractional Laplace equation: regularity of
+solutions and finite element approximations. SIAM J. Numer. Anal., 55(2):472–
+495, 2017.
+[ABH19]
+G. Acosta, J. P. Borthagaray, and N. Heuer. Finite element approximations of the
+nonhomogeneous fractional Dirichlet problem. IMA J. Numer. Anal., 39(3):1471–
+1501, 2019.
+[AGG94]
+C. Amrouche, V. Girault, and J. Giroire. Weighted Sobolev spaces for Laplace’s
+equation in Rn. J. Math. Pures Appl. (9), 73(6):579–606, 1994.
+[BBN+18] A. Bonito, J. P. Borthagaray, R. H. Nochetto, E. Ot´arola, and A. J. Salgado. Nu-
+merical methods for fractional diffusion. Comput. Vis. Sci., 19(5-6):19–46, 2018.
+[BMN+19] L. Banjai, J. M. Melenk, R. H. Nochetto, E. Ot´arola, A. J. Salgado, and C. Schwab.
+Tensor FEM for spectral fractional diffusion. Found. Comput. Math., 19(4):901–962,
+2019.
+[BV16]
+C. Bucur and E. Valdinoci. Nonlocal diffusion and applications, volume 20 of Lecture
+Notes of the Unione Matematica Italiana. Springer, [Cham]; Unione Matematica
+Italiana, Bologna, 2016.
+[Cos88]
+M. Costabel. A symmetric method for the coupling of finite elements and boundary
+elements. In The mathematics of finite elements and applications, VI (Uxbridge,
+1987), pages 281–288. Academic Press, London, 1988.
+[CS07]
+L. Caffarelli and L. Silvestre. An extension problem related to the fractional Lapla-
+cian. Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007.
+[DDG+20] M. D’Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian, and Z. Zhou. Numerical
+methods for nonlocal and fractional models. Acta Numer., 29:1–124, 2020.
+21
+
+[FKM22]
+M. Faustmann, M. Karkulik, and J. M. Melenk. Local convergence of the FEM for
+the integral fractional Laplacian. SIAM J. Numer. Anal., 60(3):1055–1082, 2022.
+[FMMS22] M. Faustmann, C. Marcati, J. M. Melenk, and C. Schwab.
+Weighted Analytic
+Regularity for the Integral Fractional Laplacian in Polygons. SIAM J. Math. Anal.,
+54(6):6323–6357, 2022.
+[FR22]
+M. Faustmann and A. Rieder. FEM-BEM coupling in fractional diffusion. Work in
+progress, 2022.
+[Han90]
+H. Han.
+A new class of variational formulations for the coupling of finite and
+boundary element methods. J. Comput. Math., 8(3):223–232, 1990.
+[KM19]
+M. Karkulik and J. M. Melenk. H-matrix approximability of inverses of discretiza-
+tions of the fractional Laplacian. Adv. Comput. Math., 45(5-6):2893–2919, 2019.
+[Kwa17]
+M. Kwa´snicki. Ten equivalent definitions of the fractional Laplace operator. Fract.
+Calc. Appl. Anal., 20(1):7–51, 2017.
+[LPG+20]
+A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai,
+M. M. Meerschaert, M. Ainsworth, and G. Karniadakis.
+What is the fractional
+Laplacian? A comparative review with new results. J. Comput. Phys., 404:109009,
+62, 2020.
+[Muc72]
+B. Muckenhoupt. Hardy’s inequality with weights. Studia Math., 44:31–38, 1972.
+[NOS15]
+R. H. Nochetto, E. Ot´arola, and A. J. Salgado. A PDE approach to fractional diffu-
+sion in general domains: a priori error analysis. Found. Comput. 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
+

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. B¨ar, Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom.
+16 (1998), no. 6, 573-596.
+[5] T. Bartsch, T. Xu, Curvature effect in the spinorial Yamabe problem on product manifolds.
+Cal. Var. PDEs. 61 (2022), no. 5, Paper No. 194, 35 pp.
+[6] W. Borrelli, Symmetric solutions for a 2D critical Dirac equation, accepted by Commun.
+Contemp. Math. 24 (2020), 2150019.
+[7] W. Borrelli, Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity,
+J. Diff. Equ. 263 (2017), pp. 7941-7964.
+[8] W. Borrelli and R. L. Frank, Sharp decay estimates for critical Dirac equations, Trans.
+Amer. Math. Soc. 373 (2020), pp. 2045-2070.
+[9] W. Borrelli, A. Malchiodi, R. Wu, Ground state dirac bubbles and Killing spinors, Comm.
+Math. Phys. 383 (2021), 1151-1180.
+[10] T. P. Branson, Second order conformal covariants, Proc. Amer. Math. Soc. 126 (1998),
+1031-1042.
+[11] J. Cuevas–Maraver, P.G. Kevrekidis, A. Saxena, A. Comech, R. Lan, Stability of soli-
+tary waves and vortices in a 2D nonlinear Dirac model, Phys. Rev. Lett. 116:21 (2016),
+214101.
+[12] A. DelaTorre, M. del Pino, M.d.M. Gonzalez, J.C. Wei, Delaunay-type singular solutions
+for the fractional Yamabe problem, Math. Ann. 369 (2017), 597-626.
+[13] H. D. Fegan, Conformally invariant first order differential operators, Quart. J. Math. Ox-
+ford, II. series 27 (1976), 371-378.
+[14] R. Finkelstein, R. LeLevier, M. Ruderman, Nonlinear Spinor Fields, Phys. Rev. 83, 326
+(1951).
+[15] F. Finster, J. Smoller, and S.-T. Yau, Particlelike solutions of the einstein-dirac equations,
+Physical Review D, 59 (1999), p. 104020.
+[16] F. Finster, J. Smoller, S.T. Yau, Particle-like solutions of the Einstein-Dirac equations,
+Physical Review D. Particles and Fields. Third Series 59 (1999) 104020
+
+39
+[17] T. Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J. Geom.
+Phys. 28:1-2 (1998), 143-157.
+[18] T. Friedrich, Dirac Operators in Riemannian Geometry, Grad. Stud. Math., vol 25, Amer.
+Math. Soc., Providence (2000).
+[19] N. Ginoux: The Dirac Spectrum. Lecture Notes in Mathematics, vol. 1976. Springer,
+Berlin (2009).
+[20] R. Gover, L. J. Peterson, Conformally invariant powers of the Laplacian, Q-curvature, and
+tractor calculus, Comm. Math. Phys. 235 (2003), 339-378.
+[21] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and
+Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.
+[22] O. Hijazi, Spectral properties of the Dirac operator and geometrical structures, in ”Geo-
+metric Methods for Quantum Field Theory”. Proceedings of the Summer School, eds. H.
+Ocampo et al.,Villa de Leyva, Colombia, July 12-30, 1999, World Scientific, Singapore,
+2001, 116-169.
+[23] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1-55.
+[24] A. Hyder, Y. Sire, Singular solutions for the constant Q-curvature problem, J. Funct. Anal.
+280:3 (2021), 108819.
+[25] K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann.
+245:2 (1979), 89-99.
+[26] B. G. Konopelchenko, Induced surfaces and their integrable dynamics, Stud. Appl. Math.
+96:1 (1996), 9-51.
+[27] R. Kusner, N. Schmitt, The spinor representation of surfaces in space, dg-ga/9610005.
+[28] H.B. Lawson, M.L. Michelson, Spin Geometry, Princeton University Press (1989).
+[29] A. Maalaoui, Infinitely many solutions for the spinorial Yamabe problem on the round
+sphere, NoDEA Nonlinear Differential Equations Appl. 23 (2016), pp. Art. 25, 14.
+[30] A. Maalaoui, Rabinowitz-Floer Homology for super-quadratic Dirac equations on spin
+manifolds. J. Fixed Point Theory Appl. 13 (2013), 175–199.
+[31] S. Matsutani, Immersion anomaly of Dirac operator on surface in R3, Rev. Math. Phys.
+11:2 (1999), 171-186.
+[32] R. Mazzeo, F. Pacard, A construction of singular solutions for a semilinear elliptic equa-
+tion using asymptotic analysis, J. Diff. Geom. 44 (1996), no. 2, 331-370.
+
+40
+[33] R. Mazzeo, F. Pacard, Constant scalar curvature metrics with isolated singularities, Duke
+Math. J. 99 (1999), no. 3, 353-418.
+[34] R. Mazzeo, N. Smale, Conformally flat metrics of constant positive scalar curvature on
+subdomains of the sphere, J. Diff. Geom. 34 (1991), no. 3, 581-621.
+[35] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian
+metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987),
+120-154, Lecture Notes in Math., 1365, Springer, Berlin, 1989
+[36] R. Schoen, The existence of weak solutions with prescribed singular behavior for a con-
+formally invariant scalar equation, Comm. Pure and Appl. Math. XLI (1988), 317-392.
+[37] R. Schoen, S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature,
+Invent. Math. 92 (1988) 47-71.
+[38] Y. Sire, T. Xu, A variational analysis of the spinorial Yamabe equation on product mani-
+folds, Ann. Sc. Norm. Super. Pisa Cl. Sci. accepted.
+[39] S. Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac equa-
+tions, Ann. Henri Poincar´e 10 (2010), pp. 1377–1393.
+[40] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D.
+1 (1970), 2766-2769.
+[41] I. A. Taimanov, Modified Novikov-Veselov equation and differential geometry of sur-
+faces, Amer. Math. Soc. Transl. (2) 179 (1997), 133-151; http://arxiv.org/dg-ga/9511005.
+[42] I. A. Taimanov, The Weierstrass representation of closed surfaces in R3, Funktsional.
+Anal. i Prilozhen. 32:4 (1998), 49-62; English transl., Funct. Anal. Appl. 32 (1998), 258-
+267.
+[43] I. A. Taimanov, The Weierstrass representation of spheres in R3, the Willmore numbers,
+and soliton spheres, Trudy Mat. Inst. 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
+[email protected]
+YANNICK SIRE
+DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY,
+3400 N. CHARLES STREET, BALTIMORE, MARYLAND 21218
+[email protected]
+TIAN XU
+CENTER FOR APPLIED MATHEMATICS, TIANJIN UNIVERSITY,
+300072, TIANJIN, CHINA
+[email protected]
+

9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1a766f6278c5bda8512d4a22af343a8f8870d1eb7bb939529744f046a964a4bb
+size 6389269

9dFLT4oBgHgl3EQfuS8F/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4d5b92364f1002ebe36e4e73fa46caa2e327461fb7e48e758e2bfe7725a388ba
+size 248828

9dFPT4oBgHgl3EQfYjT_/content/2301.13074v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e34b9b817c83c8277aff1a8f5799bca85936fd36be92fa5e4056582ead86801c
+size 1909384

9dFPT4oBgHgl3EQfYjT_/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:69641247521f321389337e17c374553883fd425f8d32f17ba3f55d3d4f9a919d
+size 250509

9tAzT4oBgHgl3EQf-_7r/content/tmp_files/2301.01943v1.pdf.txt

@@ -0,0 +1,2179 @@
+Draft version January 6, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+UOCS-IX. 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
+[email protected]
+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<Gbp<16, 5<F169M−G<6) show high ro-
+tational velocity in contrast to CMD of F172M−G vs.
+Gbp (15<Gbp<16, 4<G172M−M<5). It is fairly well
+established that high rotational velocities enhance the
+coronal and transitional line emissions (Pallavicini et al.
+1981; Linsky et al. 2020). Thus, it is consistent with the
+suggestion that high rotation stars are on the blue side
+because of high emission line activity in total contrast to
+the MS of F172M−G vs. Gbp CMD. This phenomenon
+sets into stars redder than (Gbp−Grp) ∼0.5 mag.
+5. SPECTRAL ENERGY DISTRIBUTION FITS
+It is well demonstrated in previous studies of exotic
+stellar populations, such as BSSs in OCs, that they are
+products of stellar interactions. There might be a chance
+of detecting a binary companion in the case of BSSs and
+YSSs. SEDs of such systems can be used to obtain the
+parameters of the multiple components. In this section,
+we present the multi-wavelength SEDs constructed for
+the BSSs, YSSs, and CSPN identified with UVIT to
+derive their atmospheric parameters like effective tem-
+perature (Teff), luminosity (L), and radius (R). We aim
+to probe the physical nature of these stars and probable
+
+Exotic Stellar Populations in NGC 2818
+9
+2
+3
+4
+5
+6
+7
+8
+9
+F148W 
+ G
+16
+17
+18
+19
+20
+21
+22
+23
+F148W
+MS
+BSS
+YSS
+SGB
+775 Myr,[Fe/H]=0.0 dex
+2
+3
+4
+5
+6
+7
+8
+9
+F169M 
+ G
+16
+17
+18
+19
+20
+21
+22
+F169M
+MS
+BSS
+YSS
+SGB
+775 Myr,[Fe/H]=0.0 dex
+Figure 6. FUV-optical CMDs using F148W and F169M passbands of NGC 2818 of confirmed members cross-identified using
+UVIT FUV and Gaia EDR3 catalog. The error bars (median) are shown in gray color on the left side of each panel. The rest
+of the details are the same as in Figure 5.
+Table 2. Stellar parameters obtained from best SED fit of BSSs detected with UVIT in NGC 2818. Column 1 lists the star
+ID used in the paper. Columns 2 and 3 display the RA and DEC of all the stars considered for fitting, respectively. The Teff,
+luminosities, and radii of all-stars, along with errors, are tabulated in columns 4, 5, and 6, respectively. Columns 7 and 8 lists
+the reduced χ2 value corresponding to the best fit and ratio of the number of photometric data points (
+Nfit
+Ntot ) used for the fit to
+the total number of available data points.
+Star ID
+RA (deg)
+DEC (deg)
+Teff (K)
+L
+L⊙
+R
+R⊙
+χ2
+red
+Vgf
+V gfb
+Nfit
+Ntot
+BSS1
+139.0306
+-36.59184
+11, 500 ± 250
+91.55 ± 17.54
+2.39 ± 0.22
+12.9
+12.9
+1.53
+11/12
+BSS2
+139.0279
+-36.59178
+9, 000 ± 250
+32.99 ± 6.31
+2.31 ± 0.21
+3.1
+3.1
+0.88
+12/12
+BSS3
+139.1633
+-36.43083
+8, 500+500
+−250
+52.28 ± 9.84
+3.30 ± 0.31
+4.8
+4.8
+1
+19/19
+BSS4
+139.0276
+-36.6423
+8, 750 ± 250
+20.97 ± 3.94
+1.94 ± 0.18
+4.9
+4.9
+0.91
+19/19
+hot companions, if present, by estimating their stellar
+parameters and placing them on the HR diagram. SEDs
+are generated with the observed photometric data points
+spanning a wavelength range from FUV-to-IR and fit-
+ted with selected theoretical models. We made use of
+the virtual observatory tool, VOSA (VO Sed analyzer,
+Bayo et al. 2008) for SED analysis. The details of the
+SED fitting technique are described in Rani et al. (2021).
+In addition to χ2
+red, VOSA calculates two extra parame-
+ters, Vgf and V gfb, known as modified χ2
+red to estimate
+the goodness of fit in case the observational flux errors
+are too small. The value of V gfb should be less than 15
+to achieve a reliable SED fit (Rebassa-Mansergas et al.
+2021).
+The Kurucz stellar atmospheric models are employed
+to create synthetic SEDs (Castelli et al. 1997; Castelli
+& Kurucz 2003) for BSSs and YSSs, which have ob-
+served photometric data points covering a wavelength
+range from UV to IR. The free parameters available in
+the Kurucz model are Teff, metallicity, and log g. To
+fit the observed SEDs of the stars, as mentioned earlier
+with Kurucz models, we assumed Teff, and log g as free
+parameters, and fixed the value of metallicity
+�
+Fe/H
+�
+= 0.0, close to the cluster metallicity. We adopted the
+range of Teff from 5,000-50,000 K and log g from 3.5-5
+dex in the Kurucz models.
+We combined the photo-
+metric data points of UVIT (4 passbands) with GALEX
+(2 passbands), Gaia EDR3 (3 passbands) (Gaia Col-
+laboration et al. 2018), SDSS (3 passbands), APASS (2
+passbands), 2MASS (3 passbands), and WISE (4 pass-
+bands) to generate the observed SEDs. VOSA makes use
+of Fitzpatrick reddening law (Fitzpatrick 1999; Indebe-
+
+10
+Rani et al.
+Table 3. Derived parameters of YS and MS stars from the composite SED fit. The different models used to fit the cooler (A)
+and hotter (B) components of the SEDs are presented in column 5. The rest of the columns have the same meaning as depicted
+in Table 2.
+Star ID
+RA (deg)
+Dec (deg)
+Type
+Model Used
+Teff (K)
+L
+L⊙
+R
+R⊙
+χ2
+red
+Vgf
+V gfb
+Nfit
+Ntot
+YSS1
+139.0523
+-36.57946
+A
+Kurucz
+4, 750 ± 125
+338.1 ± 63.25
+27.01 ± 2.49
+5.6
+5.6
+0.36
+20/20
+B
+Koester
+10, 250 ± 250
+7.43+3.17
+−2.36
+0.864+0.105
+−0.083
+4.3
+4.3
+0.61
+YSS2
+138.9976
+-36.58243
+A
+Kurucz
+5, 000 ± 250
+78.91 ± 15.55
+10.93 ± 1
+3.5
+3.5
+0.71
+16/16
+B
+Koester
+10, 000 ± 250
+4.72+1.76
+−1.51
+0.723+0.069
+−0.069
+2.4
+2.4
+0.81
+MS
+139.0592
+-36.60989
+A
+Kurucz
+6, 000 ± 125
+18.35 ± 3.47
+3.98 ± 0.37
+7.3
+7.2
+0.99
+18/18
+B
+Kurucz
+9, 000 ± 125
+10.79 ± 2.04
+1.36 ± 0.125
+7.3
+7.2
+0.99
+Table 4. Derived parameters of PN NGC 2818 from the best SED fit. The notation of all columns is the same as described in
+Table 2
+Star ID
+RA
+DEC
+Model Used
+Teff
+L
+L⊙
+R
+R⊙
+χ2
+red
+Vgf
+V gfb
+Nfit
+Ntot
+(deg)
+(deg)
+(K)
+PN NGC 2818
+139.0061
+-36.62707
+TMAP(Grid3)
+190, 000 ± 8080.40
+826.75 ± 225.21
+0.026 ± 0.002
+8.3
+8.3
+4.5
+6/6
+3
+4
+5
+6
+F172M 
+ G
+0.1
+0.2
+0.3
+0.4
+0.5
+Gbp 
+ Grp
+0
+50
+100
+150
+200
+250
+Vsini
+Figure 7. F172M−G vs. Gbp−Grp color-color plot of all
+stars detected with UVIT color-coded by their measured
+Vsini values.
+Stars with black color symbols do not have
+estimated Vsini values.
+touw et al. 2005) to compute the extinction in different
+passbands and correct for extinction in observed fluxes
+for the provided AV . VOSA utilizes the Markov chain
+Monte Carlo (MCMC) approach to estimate the uncer-
+tainties in the stellar atmospheric parameters obtained
+using the SED fit. We estimated the radius (R) of the
+star using the scaling relation Md =
+� R
+D
+�2, where D is
+the distance to the cluster and Md is the scaling factor.
+We conducted SED fitting analysis for four BSSs, two
+YSSs, and PN, as described in the following subsections.
+5.1. Blue Straggler Stars
+The best-fitted SEDs for all BSSs are shown in Fig-
+ure 9, where the lower panel of each SED depicts the
+fractional residual between the observed and predicted
+fluxes. The overplotted black solid line presents the syn-
+thetic Kurucz model spectrum created using the param-
+eters corresponding to the best-fit SED. The star IDs
+adopted in this work are displayed on top of each SED.
+We observe that the SEDs of all BSSs are seemed to be
+well-fitted with a single model, as the residual is close
+to zero in all SEDs. Since the observed flux errors are
+very small for all the filters used, the error bars (shown
+with black color) are smaller than the data points. We
+list their parameters corresponding to the best fit in Ta-
+ble 2. We obtain V gfb values for all BSSs to be around
+1, indicating the good SED fits, and all the derived fun-
+damental parameters are also reliable. The BSSs have a
+Teff range of 8,500−11,500 K, and radii of 1.9−3.3 R⊙.
+Now, here arises the two possibilities about the nature
+of these stars: 1) either all BSSs are single stars, 2) or
+they are binaries with a very faint companion, not able
+to detect by the UVIT observations. If these stars are
+single, they are likely to be formed via the merger of the
+component stars in a binary.
+
+Exotic Stellar Populations in NGC 2818
+11
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+Gbp 
+ Grp
+12
+13
+14
+15
+16
+Gbp
+BSS
+YSS
+Vsini
+775 Myr,[Fe/H]=0.0 dex
+0
+50
+100
+150
+200
+250
+Vsini
+2
+3
+4
+5
+6
+7
+8
+F172M 
+ G
+12
+13
+14
+15
+16
+Gbp
+BSS
+YSS
+775 Myr,[Fe/H]=0.0 dex
+0
+50
+100
+150
+200
+250
+Vsini
+2
+3
+4
+5
+6
+7
+8
+9
+F169M 
+ G
+12
+13
+14
+15
+16
+Gbp
+BSS
+YSS
+775 Myr,[Fe/H]=0.0 dex
+0
+50
+100
+150
+200
+250
+Vsini
+0.4
+0.0
+0.4
+0.8
+1.2
+1.6
+F169M 
+ F172M
+12
+13
+14
+15
+16
+Gbp
+BSS
+YSS
+775 Myr,[Fe/H]=0.0 dex
+0
+50
+100
+150
+200
+250
+Vsini
+Figure 8. Optical (upper left), F172M-G vs Gbp (upper right), F169M-G vs. Gbp (lower left), and F169M-F172M vs. Gbp
+(lower right) CMDs of NGC 2818 members color-coded by measured Vsini values. The rest of the details are the same as in
+Figure 5.
+5.2. Yellow Straggler Stars
+Figure 10 presents the SEDs of two stars classi-
+fied as YSSs in this work.
+In this figure, the lower
+panel represents the fractional residual, i.e., the ratio
+of the difference between the observed and model flux
+(Fobs−Fmodel) and the observed flux at every given data
+point. We can see in Figure 10 that both YSSs are show-
+ing significant UV excess as a single model could not fit
+the entire SED. It can also be noticed in the fractional
+residual plot showing a rise in flux in the UV wave-
+lengths for a single spectrum fit (shown as an orange
+dash-dotted line in the figure). To fit the hotter compo-
+nent of the system, first, we gave excess for wavelength
+less than 3000 Å and fitted the cooler component that
+includes the optical and IR data points with the Kurucz
+model by selecting Teff range from 3,500−50,000 K and
+logg from 1.5−2.5 dex. From the single fit, the computed
+values of Teff of the YSS1 and YSS2 are 4,750 K and
+5,000 K, respectively. The radius of YSS1 and YSS2 is
+27 R⊙ and ∼ 11 R⊙, respectively. From their tempera-
+ture and radii, we infer that they are in the giant phase
+of stellar evolution. After obtaining the stellar parame-
+ters of the cooler component, then we used Binary SED
+Fitting4 code to fit the hotter part of the SED. The full
+details of this code are well described in Jadhav et al.
+(2021). As we expect the hotter component to be com-
+pact, we have used the Koester WD model (Tremblay &
+Bergeron 2009; Koester 2010). In this model, the range
+of free parameters Teff and logg is 5,000−80,000 K and
+6.5−9.5, respectively.
+The double fit of both stars is
+shown in Figure 10, where the Kurucz model fit is shown
+with an orange dash-dotted line, and the Koester model
+fit with a light-blue dashed line. The composite fit is
+marked with a solid green line.
+The fractional resid-
+ual in both plots is close to zero for all observed data
+points indicating how well the double component fit re-
+produces the observed SED. This is even evident from
+the vgfb values (close to 1) computed from the SED fit-
+ting of both stars. The estimated parameters of both
+4 https://github.com/jikrant3/Binary_SED_Fitting
+
+12
+Rani et al.
+10
+16
+10
+15
+10
+14
+10
+13
+F  [ergs/s/cm2/A]
+BSS1       [Fe/H] = 0.0, Temperature = 11,500 K
+model spectrum
+Model Flux
+Observed
+No Fit
+1200
+2000
+3000
+5000
+10000
+25000
+[Å]
+0.5
+0.0
+0.5
+Fractional
+Residual
+10
+17
+10
+16
+10
+15
+10
+14
+F  [ergs/s/cm2/A]
+BSS2       [Fe/H] = 0.0, Temperature = 9,000 K
+model spectrum
+Model Flux
+Observed
+1200
+2000 3000
+5000
+10000
+25000
+50000
+[Å]
+0.5
+0.0
+0.5
+Fractional
+Residual
+10
+17
+10
+16
+10
+15
+10
+14
+10
+13
+F  [ergs/s/cm2/A]
+BSS3       [Fe/H] = 0.0, Temperature = 8,500 K
+model spectrum
+Model Flux
+Observed
+1200
+2000 3000
+5000
+10000
+25000
+50000
+[Å]
+0.5
+0.0
+0.5
+Fractional
+Residual
+10
+17
+10
+16
+10
+15
+10
+14
+F  [ergs/s/cm2/A]
+BSS4       [Fe/H] = 0.0, Temperature = 8,750 K
+model spectrum
+Model Flux
+Observed
+1200
+2000 3000
+5000
+10000
+25000
+50000
+[Å]
+0.5
+0.0
+0.5
+Fractional
+Residual
+Figure 9. SEDs of four BSSs detected with UVIT. Extinction correction has been incorporated in all the observed photometric
+fluxes from UV to IR. The BSS ID adopted in this work is shown in each figure. The gray color presents the best-fitting Kurucz
+model spectrum in all the plots. The data points that are excluded in the SED fit are shown with the orange color-filled symbol.
+The bottom panel in all the SEDs illustrates the residual between the observed fluxes and model predictions.
+YSSs from the best binary fit are tabulated in Table 3.
+From the double fit, we estimate the Teff of the hotter
+companion of YSS1 and YSS2 are 10,250 K and 10,000
+K, respectively. The values of parameters such as Teff,
+luminosities, and radii of the stars are mentioned on the
+top of each SED.
+5.3. PN NGC 2818
+As we have shown in the previous section, the PN
+NGC 2818 most likely has a physical association with
+the cluster; it will be interesting to characterize its cen-
+tral star to obtain information about its progenitor. We
+can clearly see the CSPN in the FUV image, as shown
+in Figure 1, implying its very high temperature. The
+magnitude of CSPN is a vital parameter to study its
+evolution as it can be used to determine the stellar pa-
+rameters. The magnitude of the CSPN in optical filters
+was measured by Gathier & Pottasch (1988). As CSPN
+is well observed in all FUV images, therefore we have
+calculated the magnitude of the central star by perform-
+ing the PSF photometry on the FUV images acquired
+in 1st and 2nd epoch observations. We have subtracted
+the nebular background in assessing the magnitude of
+the CSPN. The external extinction and distance to the
+nebula are considered to be the same as that of the
+cluster.
+Four FUV UVIT data points are combined
+with two optical photometric data points from Gathier
+& Pottasch (1988) to construct the observed SED of the
+nebula. As the central star seemed to be very hot, we
+have fitted its SED with the Tübingen NLTE Model At-
+mosphere Package (TMAP) (Grid3) model used for hot
+stars (Rauch & Deetjen 2003; Werner et al. 2003). This
+model grid spans a range of atmospheric parameters
+such as 50, 000K ≤ Teff ≤ 190, 000K, 5.0 ≤ logg ≤ 9.0,
+and 0 ≤ XH ≤ 1.
+It is important to note that we
+took into account the external extinction while fitting
+its SED but did not incorporate the internal extinction
+in the nebula. We have noticed that Teff derived using
+the TMAP model fit to the observed SED corresponds
+to their upper limit, which indicates that this star is
+likely to be hotter than the estimated temperature from
+this model. The stellar parameters computed from the
+best-fit SED of the nebula are summarised in Table 4.
+
+Exotic Stellar Populations in NGC 2818
+13
+Figure 10. Double-fit SEDs of YSSs. The meaning of all the symbols is displayed in the legend. The star IDs and parameters
+of two components obtained from the fit are shown on the top of both SEDs. The green color represents the composite model
+flux along with the observed fluxes marked with red symbols. Orange dotted-dash and blue dashed lines indicate Kurucz and
+Koester models used to fit the star’s cooler and hotter components, respectively. The middle panel presents the fractional
+residual (Orange dashed line) corresponding to the single fit as well as the composite fit (Green solid line). The fractional
+observational uncertainties in the flux are also shown here. The values of χ2
+red and modified χ2
+red parameter, namely vgf 2
+b
+representing the best-fit are displayed in the lower panel.
+
+YSS1
+A (4750.0 K, logg=3.5)
++
+Model
+Obs
+ Model
+B
+10-13
+++ +
+10-14
+ +
+10-15
+10-16
+10-17
+10-18
+1.0
+Residual
+0.5
+0.0
+104
+105
+Wavelength (A)YSS2
+B (1000058 K, 4.715271:35
+A (5000.0 K, logg=1.5)
+10-13
+Obs
+Model
+口
+Model
+A
+B
+10-14
+7
+10-15
+10-16
+10-17
+10-18
+1
+Fractional
+104
+105
+Wavelength (A)14
+Rani et al.
+10
+16
+10
+15
+10
+14
+F  [ergs/s/cm2/A]
+PN NGC 2818       Temperature = 190,000 K
+Model
+Model Flux
+UVIT
+1200
+2000
+3000
+5000
+7000
+[Å]
+0.5
+0.0
+0.5
+Fractional
+Residual
+10
+18
+10
+17
+10
+16
+10
+15
+10
+14
+F  [ergs/s/cm2/A]
+MS       A (6000 K, logg=4.0)       B (9000 K, logg=4.5)
+Model
+A
+B
+Model
+Obs
+104
+[Å]
+0
+1
+Fractional
+Residual
+Figure 11. SED fit of the CSPN (left panel) and MS star (right panel) after taking into account the extinction correction. The
+black solid line represents the theoretical TMAP model fit to the observed fluxes shown with red symbols. The best-fit Teff
+value is displayed in the figure. The rest of the details are the same as in Figure 9 and 10.
+5.4. MS stars
+We also have constructed the SEDs for the MS stars
+detected with UVIT, for which rotational velocity in-
+formation was available in the literature to investigate
+their nature.
+Apart from that, we also have consid-
+ered the MS stars for SED analysis for which rotational
+velocity was not estimated earlier, and their position
+in all FUV-optical CMDs was not matched with their
+expected one. 31 MS stars with the known rotational
+velocity are identified with UVIT in two epochs. Other
+than these stars, 6 MS stars are brighter than MS turn-
+off in FUV CMDs. We have used the Kurucz models
+to fit their observed SEDs to obtain their physical pa-
+rameters and check their binarity. Out of 37 stars, we
+observed that only one MS star shows significant FUV
+excess, as displayed in the right panel of Figure 11,
+whereas other stars show less or mild UV excess that
+could not be fitted with a double component SED. Chro-
+mospheric activity in the above star cannot account for
+UV excess as it is exceptionally high compared to the
+model. The other possibility to explain this excess is
+the presence of a hot companion that mainly emits at
+shorter wavelengths.
+To account for the presence of
+the hot companion, we fitted the entire SED with the
+Kurucz model using the binary fit task from VOSA.
+The double component fit for this star is found to be
+satisfactory (Right panel of Figure 11), and the best-fit
+parameters computed are tabulated in Table 3.
+The
+radii of both components suggest that they are not
+quite on the MS. The cooler companion is likely to be
+a sub-giant (R/R⊙ ∼ 4.0), whereas the hot companion
+has a smaller radius (R/R⊙ ∼ 1.36) when compared
+to the MS star of similar temperature (R/R⊙ ∼ 6.0).
+It might be possible that this is a post-mass transfer
+system where the hotter component is the donor, and
+the cooler component is still bloated after gaining mass.
+The rotational velocity (Vsini) of this star is around 39
+km/s.
+6. EVOLUTIONARY STATUS
+Placing the stars on the HR diagram provides informa-
+tion about their evolutionary stage and helps in probing
+the nature of the hot companions in the case of binary
+stars. To examine the evolutionary status of exotic stars
+considered in this study, we have plotted the theoreti-
+cal evolutionary sequences starting from the MS to the
+moment the star has entered the tip of the RGB stage.
+These tracks are taken from MIST models computed by
+Choi et al. (2016); Paxton et al. (2018) and selected for
+the cluster age and metallicity close to the cluster metal-
+licity. The stellar parameters estimated from the single
+SED fit for four BSSs are plotted in the HR diagram.
+The meaning of the color and symbols are marked in
+Figure 12. We can notice in Figure 12 that BSSs are
+lying bluer to the MS track, suggesting that these four
+stars belong to the BS evolutionary phase.
+The location of two YSSs on the HR diagram is near
+the theoretical RGB sequence. It indicates that their
+progenitors’ (BSSs) have already evolved into a giant
+phase where the contracting helium core is surrounded
+by the hydrogen-burning shell. The hot companions of
+both YSSs seemed to be compact in nature, as indicated
+by their estimated radii suggesting they might belong to
+the WD or extremely low mass (ELM) WD or subdwarf
+stage of stellar evolution. In addition to the MS tracks,
+we have presented the DA-type WD cooling sequences
+with masses 0.5M⊙ and 0.2M⊙ taken from Tremblay
+et al. (2011) in Figure 12. From comparing the position
+
+Exotic Stellar Populations in NGC 2818
+15
+of the hot companions of both YSSs with theoretical
+WD cooling tracks, we notice that their location is not
+reproduced by them, implying that they still have not
+entered the WD stage. While there are non-DA type
+WDs that are believed to result from mergers, they are
+not expected to be found in OCs because the merger
+process would take longer than the age of the cluster.
+In order to find out where ELM WDs fall in the HR
+diagram, we have used the field ELM WD catalog pro-
+vided by Brown et al. (2016). They have estimated the
+T eff and log g values of the considered ELM WD sample
+in their paper. To place them on the Teff vs. luminosity
+plot, SED fitting technique is used to estimate the lumi-
+nosity of all ELM WDs (Priv. Comm. Vikrant Jadhav).
+The extinction correction has been incorporated in all
+the stars. All field ELM WDs are marked as cyan-filled
+symbols in Figure 12. We note that the hot companions
+of the YSSs are more luminous than the field ELMs with
+a similar temperature.
+As the location of the binary companions of YSSs is
+not reproduced by the WD tracks as well as ELM WDs,
+we further suspect that they might belong to the class of
+A-type subdwarfs (sdA) as they are lying near the gen-
+eral location of subdwarfs in the HR diagram. sdA stars
+are supposed to occupy the location between the dwarfs
+and WDs in the HR diagram; hence, they are more com-
+pact than dwarfs, indicating a higher log g value. Brown
+et al. (2017) performed a detailed study of sdA stars to
+investigate their physical nature and a possible link to
+the ELM WDs. We used the field sdA catalog to locate
+their positions on the HR diagram.
+As only effective
+temperatures of all sdA stars were available in the cata-
+log, we used the SED fitting technique to determine their
+luminosities. The extinction in the visual band (AV ) for
+these stars was estimated using the reddening map pro-
+vided by Schlafly & Finkbeiner (2011). We have taken
+care of the extinction correction in the observed fluxes
+in different bands of all sdA stars.
+The distances to
+these stars are available in the Gaia EDR3 catalog. We
+have used the distances reported in Bailer-Jones et al.
+(2021), estimated using Gaia EDR3 catalog, and they
+all fall within a range of ∼1.5 to 8 kpc. The sdA stars
+are displayed with purple-filled symbols in the HR di-
+agram. The hot companions of YSSs are found to be
+hotter than the similarly luminous field sdAs and more
+luminous than the similarly hot field sdAs.
+From this comparison, we suggest that they are most
+likely to be sdA stars formed through a binary mass
+transfer scenario. These binaries are probably a post-
+mass-transfer system consisting of an A-type subdwarf
+candidate and a YS star. We also checked the position
+of the hotter and cooler components of the MS star on
+the HR diagram displayed with orange-color symbols.
+The hotter component occupies a location bluer than
+theoretical isochrone, might be evolving to the sdA type
+star, whereas the cooler component occupies the loca-
+tion expected for sub-giants. The evolution of this star
+might be similar to the YSS as the cooler component is
+evolving to the giant stage, whereas the hotter compo-
+nent later might end up as sdA. Thus, we speculate that
+this system might be a progenitor of the YSSs detected
+in this cluster.
+Further, we have used the pAGB models computed
+by Miller Bertolami (2016) to deduce the evolutionary
+state of the CSPN. We adopted the cluster metallicity
+(Z=∼ 0.02 dex) to select the pAGB tracks. Tracks with
+a range of final mass as shown in Figure 12 are presented
+from the beginning of the pAGB phase when the H-rich
+envelope drops below Menv = 0.01M∗ to the moment
+the star has already entered its WD cooling sequence
+at L∗ = Lsun.
+The estimated parameters of the PN
+from the SED fit are plotted in the HR diagram (Red
+filled symbol). From the comparison to these theoretical
+pAGB tracks, we observe that CSPN is found to be lo-
+cated on the track (Black dash-dotted line) correspond-
+ing to the final mass 0.657Msun. It can be noted from
+here that the star has already entered the WD cooling
+phase.
+7. DISCUSSION
+We have conducted an observational study of OC
+NGC 2818 and the PN within its field using FUV
+medium-resolution
+space-based
+imaging
+data
+from
+UVIT aboard AstroSat. This paper aims to use the most
+accurate and complete Gaia EDR3 data on stellar as-
+trometry and photometry in the nearby intermediate age
+OC NGC 2818 to establish the membership probability
+of known stars and to deduce the evolutionary state of
+exotic stars. Since the stars reside in the central area of
+the cluster, we have confined ourselves with the consid-
+eration of the inner part of the cluster with a radius of
+30′ and selected 37508 stars brighter than G = 21 mag.
+Using the GMM method to pick out the PM members,
+we have chosen 718 stars as the cluster members with
+Pµ > 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. This research also made
+use of the Aladin sky atlas developed at CDS, Stras-
+bourg Observatory, France (Bonnarel et al. 2000).
+Software: GaiaTools (Vasiliev 2019), Topcat (Taylor
+2011), Matplotlib (Hunter 2007), NumPy (van der Walt
+et al. 2011), Scipy (Oliphant 2007; Millman & Aivazis
+2011), Astropy (Astropy Collaboration et al. 2013, 2018)
+and Pandas (McKinney 2010)
+REFERENCES
+Ahumada, J. A., & Lapasset, E. 2007, A&A, 463, 789,
+doi: 10.1051/0004-6361:20054590
+Alam, S., Albareti, F. D., Allende Prieto, C., et al. 2015,
+ApJS, 219, 12, doi: 10.1088/0067-0049/219/1/12
+Astropy Collaboration, Robitaille, T. P., Tollerud, E. J.,
+et al. 2013, A&A, 558, A33,
+doi: 10.1051/0004-6361/201322068
+Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M.,
+et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
+Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M.,
+Demleitner, M., & Andrae, R. 2021, VizieR Online Data
+Catalog, I/352
+Bastian, N., & de Mink, S. E. 2009, MNRAS, 398, L11,
+doi: 10.1111/j.1745-3933.2009.00696.x
+Bastian, N., Kamann, S., Cabrera-Ziri, I., et al. 2018,
+MNRAS, 480, 3739, doi: 10.1093/mnras/sty2100
+Bayo, A., Rodrigo, C., Barrado Y Navascués, D., et al.
+2008, A&A, 492, 277, doi: 10.1051/0004-6361:200810395
+Bianchi, L., Shiao, B., & Thilker, D. 2017, ApJS, 230, 24,
+doi: 10.3847/1538-4365/aa7053
+Boffin, H. M. J., Carraro, G., & Beccari, G., eds. 2015,
+Astrophysics and Space Science Library, Vol. 413,
+Ecology of Blue Straggler Stars (Springer Berlin
+Heidelberg), doi: 10.1007/978-3-662-44434-4
+Bohigas, J. 2003, RMxAA, 39, 149
+—. 2008, ApJ, 674, 954, doi: 10.1086/524977
+Bonnarel, F., Fernique, P., Bienaymé, O., et al. 2000,
+A&AS, 143, 33
+Brandt, T. D., & Huang, C. X. 2015, ApJ, 807, 24,
+doi: 10.1088/0004-637X/807/1/24
+Brown, W. R., Gianninas, A., Kilic, M., Kenyon, S. J., &
+Allende Prieto, C. 2016, ApJ, 818, 155,
+doi: 10.3847/0004-637X/818/2/155
+Brown, W. R., Kilic, M., & Gianninas, A. 2017, ApJ, 839,
+23, doi: 10.3847/1538-4357/aa67e4
+Cabrera-Ziri, I., Bastian, N., Hilker, M., et al. 2016,
+MNRAS, 457, 809, doi: 10.1093/mnras/stv2977
+Cantat-Gaudin, T., & Anders, F. 2020, A&A, 633, A99,
+doi: 10.1051/0004-6361/201936691
+Cantat-Gaudin, T., Anders, F., Castro-Ginard, A., et al.
+2020, A&A, 640, A1, doi: 10.1051/0004-6361/202038192
+Castelli, F., Gratton, R. G., & Kurucz, R. L. 1997, A&A,
+318, 841
+Castelli, F., & Kurucz, R. L. 2003, in IAU Symposium, Vol.
+210, Modelling of Stellar Atmospheres, ed. N. Piskunov,
+W. W. Weiss, & D. F. Gray, A20.
+https://arxiv.org/abs/astro-ph/0405087
+Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102,
+doi: 10.3847/0004-637X/823/2/102
+Cummings, J. D., Kalirai, J. S., Tremblay, P. E.,
+Ramirez-Ruiz, E., & Choi, J. 2018, ApJ, 866, 21,
+doi: 10.3847/1538-4357/aadfd6
+Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003,
+VizieR Online Data Catalog, II/246
+Cutri, R. M., Wright, E. L., Conrow, T., et al. 2021, VizieR
+Online Data Catalog, II/328
+de Marchi, F., de Angeli, F., Piotto, G., Carraro, G., &
+Davies, M. B. 2006, A&A, 459, 489,
+doi: 10.1051/0004-6361:20064898
+Dufour, R. J. 1984, ApJ, 287, 341, doi: 10.1086/162694
+Ferraro, F. R., Lanzoni, B., Dalessandro, E., et al. 2012,
+Nature, 492, 393, doi: 10.1038/nature11686
+Fitzpatrick, E. L. 1999, PASP, 111, 63, doi: 10.1086/316293
+Frew, D. J., Parker, Q. A., & Bojičić, I. S. 2016, MNRAS,
+455, 1459, doi: 10.1093/mnras/stv1516
+
+Exotic Stellar Populations in NGC 2818
+19
+Gaia Collaboration, Helmi, A., van Leeuwen, F., et al.
+2018, A&A, 616, A12, doi: 10.1051/0004-6361/201832698
+Gathier, R., & Pottasch, S. R. 1988, A&A, 197, 266
+Gosnell, N. M., Mathieu, R. D., Geller, A. M., et al. 2015,
+ApJ, 814, 163, doi: 10.1088/0004-637X/814/2/163
+Gossage, S., Conroy, C., Dotter, A., et al. 2019, ApJ, 887,
+199, doi: 10.3847/1538-4357/ab5717
+Henden, A. A., Levine, S., Terrell, D., & Welch, D. L. 2015,
+in American Astronomical Society Meeting Abstracts,
+Vol. 225, American Astronomical Society Meeting
+Abstracts #225, 336.16
+Hills, J. G., & Day, C. A. 1976, Astrophys. Lett., 17, 87
+Hunter, J. D. 2007, Computing in Science and Engineering,
+9, 90, doi: 10.1109/MCSE.2007.55
+Iben, I., J., & Tutukov, A. V. 1999, in Astronomical Society
+of the Pacific Conference Series, Vol. 169, 11th European
+Workshop on White Dwarfs, ed. S. E. Solheim & E. G.
+Meistas, 432
+Indebetouw, R., Mathis, J. S., Babler, B. L., et al. 2005,
+ApJ, 619, 931, doi: 10.1086/426679
+Jadhav, V. V., Pandey, S., Subramaniam, A., & Sagar, R.
+2021, Journal of Astrophysics and Astronomy, 42, 89,
+doi: 10.1007/s12036-021-09746-y
+Jadhav, V. V., Sindhu, N., & Subramaniam, A. 2019, ApJ,
+886, 13, doi: 10.3847/1538-4357/ab4b43
+Jadhav, V. V., & Subramaniam, A. 2021, MNRAS, 507,
+1699, doi: 10.1093/mnras/stab2264
+Jordan, C., & Linsky, J. L. 1987, in Astrophysics and Space
+Science Library, Vol. 129, Exploring the Universe with
+the IUE Satellite, ed. Y. Kondo & W. Wamsteker, 259,
+doi: 10.1007/978-94-009-3753-6_12
+Kharchenko, N. V., Piskunov, A. E., Schilbach, E., Röser,
+S., & Scholz, R. D. 2013, A&A, 558, A53,
+doi: 10.1051/0004-6361/201322302
+Knigge, C., Dieball, A., Maíz Apellániz, J., et al. 2008,
+ApJ, 683, 1006, doi: 10.1086/589987
+Koester, D. 2010, Mem. Soc. Astron. Italiana, 81, 921
+Kohoutek, L., Roth-Hoeppner, M. L., & Laustsen, S. 1986,
+A&A, 162, 232
+Kwitter, K. B., Méndez, R. H., Peña, M., et al. 2014,
+RMxAA, 50, 203. https://arxiv.org/abs/1403.2246
+Landsman, W., Aparicio, J., Bergeron, P., Di Stefano, R., &
+Stecher, T. P. 1997, ApJL, 481, L93, doi: 10.1086/310654
+Lindegren, L., Hernández, J., Bombrun, A., et al. 2018,
+A&A, 616, A2, doi: 10.1051/0004-6361/201832727
+Linsky, J. L., & Haisch, B. M. 1979, ApJL, 229, L27,
+doi: 10.1086/182924
+Linsky, J. L., Wood, B. E., Youngblood, A., et al. 2020,
+ApJ, 902, 3, doi: 10.3847/1538-4357/abb36f
+Mata, H., Ramos-Larios, G., Guerrero, M. A., et al. 2016,
+MNRAS, 459, 841, doi: 10.1093/mnras/stw646
+McCrea, W. H. 1964, MNRAS, 128, 147,
+doi: 10.1093/mnras/128.2.147
+McKinney, W. 2010, in Proceedings of the 9th Python in
+Science Conference, ed. S. van der Walt & J. Millman, 51
+– 56, doi: 10.25080/Majora-92bf1922-00a
+Meatheringham, S. J., Wood, P. R., & Faulkner, D. J. 1988,
+ApJ, 334, 862, doi: 10.1086/166882
+Mermilliod, J. C., Clariá, J. J., Andersen, J., Piatti, A. E.,
+& Mayor, M. 2001, A&A, 375, 30,
+doi: 10.1051/0004-6361:20010845
+Miller Bertolami, M. M. 2016, A&A, 588, A25,
+doi: 10.1051/0004-6361/201526577
+Millman, K., & Aivazis, M. 2011, Computing in Science &
+Engineering, 13, 9 , doi: 10.1109/MCSE.2011.36
+Milne, D. K., & Aller, L. H. 1975, A&A, 38, 183
+Niederhofer, F., Georgy, C., Bastian, N., & Ekström, S.
+2015, MNRAS, 453, 2070, doi: 10.1093/mnras/stv1791
+Oliphant, T. E. 2007, Computing in Science and
+Engineering, 9, 10, doi: 10.1109/MCSE.2007.58
+Pallavicini, R., Golub, L., Rosner, R., et al. 1981, ApJ, 248,
+279, doi: 10.1086/159152
+Panthi, A., Vaidya, K., Jadhav, V., et al. 2022, MNRAS,
+516, 5318, doi: 10.1093/mnras/stac2421
+Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS,
+234, 34, doi: 10.3847/1538-4365/aaa5a8
+Pedreros, M. 1989, AJ, 98, 2146, doi: 10.1086/115284
+Perets, H. B., & Fabrycky, D. C. 2009, ApJ, 697, 1048,
+doi: 10.1088/0004-637X/697/2/1048
+Postma, J. E., & Leahy, D. 2017, PASP, 129, 115002,
+doi: 10.1088/1538-3873/aa8800
+Pottasch, S. R. 1984, Planetary nebulae. A study of late
+stages of stellar evolution, Vol. 107 (Springer
+Netherlands), doi: 10.1007/978-94-009-7233-9
+Rain, M. J., Ahumada, J. A., & Carraro, G. 2021, A&A,
+650, A67, doi: 10.1051/0004-6361/202040072
+Rani, S., Pandey, G., Subramaniam, A., et al. 2021, ApJ,
+923, 162, doi: 10.3847/1538-4357/ac2eb6
+Rauch, T., & Deetjen, J. L. 2003, in Astronomical Society
+of the Pacific Conference Series, Vol. 288, Stellar
+Atmosphere Modeling, ed. I. Hubeny, D. Mihalas, &
+K. Werner, 103. https://arxiv.org/abs/astro-ph/0403239
+Rebassa-Mansergas, A., Solano, E., Jiménez-Esteban,
+F. M., et al. 2021, MNRAS, 506, 5201,
+doi: 10.1093/mnras/stab2039
+Riello, M., De Angeli, F., Evans, D. W., et al. 2021, A&A,
+649, A3, doi: 10.1051/0004-6361/202039587
+
+20
+Rani et al.
+Santana, F. A., Muñoz, R. R., Geha, M., et al. 2012, in
+Astronomical Society of the Pacific Conference Series,
+Vol. 458, Galactic Archaeology: Near-Field Cosmology
+and the Formation of the Milky Way, ed. W. Aoki,
+M. Ishigaki, T. Suda, T. Tsujimoto, & N. Arimoto, 339
+Santucci, R. M., Placco, V. M., Rossi, S., et al. 2015, ApJ,
+801, 116, doi: 10.1088/0004-637X/801/2/116
+Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103,
+doi: 10.1088/0004-637X/737/2/103
+Sindhu, N., Subramaniam, A., & Radha, C. A. 2018,
+MNRAS, 481, 226, doi: 10.1093/mnras/sty2283
+Stetson, P. B. 1987, PASP, 99, 191, doi: 10.1086/131977
+Sun, W., de Grijs, R., Deng, L., & Albrow, M. D. 2021,
+MNRAS, 502, 4350, doi: 10.1093/mnras/stab347
+Surendiranath, R., Kameswara Rao, N., Sagar, R., Nathan,
+J. S., & Ghosh, K. K. 1990, Journal of Astrophysics and
+Astronomy, 11, 151, doi: 10.1007/BF02715014
+Tandon, S. N., Hutchings, J. B., Ghosh, S. K., et al. 2017,
+Journal of Astrophysics and Astronomy, 38, 28,
+doi: 10.1007/s12036-017-9445-x
+Tandon, S. N., Postma, J., Joseph, P., et al. 2020, AJ, 159,
+158, doi: 10.3847/1538-3881/ab72a3
+Taylor, M. 2011, TOPCAT: Tool for OPerations on
+Catalogues And Tables. http://ascl.net/1101.010
+Tifft, W. G., Connolly, L. P., & Webb, D. F. 1972,
+MNRAS, 158, 47, doi: 10.1093/mnras/158.1.47
+Tremblay, P. E., & Bergeron, P. 2009, ApJ, 696, 1755,
+doi: 10.1088/0004-637X/696/2/1755
+Tremblay, P. E., Bergeron, P., & Gianninas, A. 2011, ApJ,
+730, 128, doi: 10.1088/0004-637X/730/2/128
+Vaidya, K., Panthi, A., Agarwal, M., et al. 2022, MNRAS,
+511, 2274, doi: 10.1093/mnras/stac207
+van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011,
+Computing in Science and Engineering, 13, 22,
+doi: 10.1109/MCSE.2011.37
+Vasiliev, E. 2019, MNRAS, 484, 2832,
+doi: 10.1093/mnras/stz171
+Vázquez, R. 2012, ApJ, 751, 116,
+doi: 10.1088/0004-637X/751/2/116
+Weidemann, V. 2000, A&A, 363, 647
+Werner, K., Deetjen, J. L., Dreizler, S., et al. 2003, in
+Astronomical Society of the Pacific Conference Series,
+Vol. 288, Stellar Atmosphere Modeling, ed. I. Hubeny,
+D. Mihalas, & K. Werner, 31.
+https://arxiv.org/abs/astro-ph/0209535
+Whitford, A. E. 1958, AJ, 63, 201, doi: 10.1086/107725
+Zhang, C. Y. 1995, ApJS, 98, 659, doi: 10.1086/192173
+

ANAyT4oBgHgl3EQfq_kk/content/tmp_files/2301.00551v1.pdf.txt

@@ -0,0 +1,2143 @@
+arXiv:2301.00551v1  [math.PR]  2 Jan 2023
+BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND
+INHOMOGENEOUS MAGNETIC IMPURITIES
+ISAO SAUZEDDE
+Abstract. 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.
+Contents
+1.
+Introduction
+1
+1.1.
+Stochastic Green’s formula
+1
+1.2.
+Magnetic impurities
+2
+2.
+Notations
+3
+2.1.
+Differential forms and integrals
+3
+2.2.
+Winding
+4
+2.3.
+Cauchy variables
+4
+3.
+Former results
+5
+4.
+Stokes formula
+6
+4.1.
+Existence of a limit
+6
+4.2.
+Strategy for the Stokes’ formula
+7
+4.3.
+Additivity
+8
+4.4.
+Contribution from the small loops
+9
+4.5.
+Stratonovich integral as a limit of integrals along piecewise-linear paths
+12
+5.
+Magnetic impurities
+14
+6.
+Funding
+19
+References
+19
+1. Introduction
+1.1. Stochastic Green’s formula. 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. 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. 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.
+When the smooth loop is replaced with a Brownian one, such a formula cannot be written down
+directly. For its left-hand side, we do have a genuine candidate provided by the Stratonovich
+integrale of η along X. 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.
+In such a framework, the Green formula is thus a convergence result rather than an equality.
+University of Warwick
+E-mail address: [email protected].
+2020 Mathematics Subject Classification. Primary 60J65; 60K37 Secondary 60G17.
+1
+
+2
+BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. 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.
+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.
+The first goal of this paper is to extend such a formula to any differential 1-form η with
+regularity C1+ǫ.
+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).
+Theorem 1. 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. Then, almost surely, for all ǫ > 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} =
+�
+−∞<k≤−n
+AX
+k .
+We also write AX
+k (resp. DX
+k ) for the Lebesgue measure of AX
+k (resp. DX
+k ), and we drop the
+superscript X when there is no doubt about the curve we are talking about.
+For a real number z and a positive integer n, we set
+[x]n =
+���
+
+
+−n
+if x ≤ −n,
+x
+if − n ≤ x ≤ n,
+n
+if x ≥ n.
+Once we have shown that the limit
+lim
+k→∞
+�
+R2 f(z)[nX(z)]kdz
+almost surely exists for all f ∈ Cǫ(R2), we will write −�
+R2 f(z)nX(z)dz for this limit.
+For a locally finite set of points P, we define nX(P) as the sum �
+z∈P nX(z). If we are also
+given a function f on R2, we define nX(P, f) as the weighted sum
+nX(P, f) =
+�
+z∈P
+nX(z)f(z).
+2.3. Cauchy variables. The Cauchy distribution C(p, σ) with position parameter p and scale
+parameter σ > 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. Some theorems on stable processes. Transactions of the American
+Mathematical Society, 95:263–273, 1960.
+[2] Sudip Chakravarty and Albert Schmid. Weak localization: The quasiclassical theory of electrons in a random
+potential. Physics Reports, 140(4):193–236, 1986.
+[3] Robert Chen and Larry A. Shepp. On the sum of symmetric random variables. Amer. Statist., 37(3):237,
+1983.
+[4] Jean Desbois, Cyril Furtlehner, and Stéphane Ouvry. Random Magnetic Impurities and the delta Impurity
+Problem. Journal de Physique I, 6:641–648, 1996. 13 pages, latex, 1 figure upon request.
+[5] Jean Luc Desbois, Cyril Furtlehner, and Stéphane Ouvry. Random magnetic impurities and the landau
+problem. Nuclear Physics, 453:759–776, 1995.
+[6] Oliver Johnson and Richard Samworth. Central limit theorem and convergence to stable laws in Mallows
+distance. Bernoulli, 11(5):829–845, 2005.
+[7] Niclas Lindvall, Abhay Shivayogimath, and A. Yurgens. Measurements of weak localization of graphene in
+inhomogeneous magnetic fields. JETP Letters, 102:367–371, 09 2015.
+[8] J. Rammer and A. L. Shelankov. Weak localization in inhomogeneous magnetic fields. Phys. Rev. B, 36:3135–
+3146, Aug 1987.
+[9] Isao Sauzedde. Planar brownian motion winds evenly along its trajectory, 2021. arXiv:2102.12372.
+[10] Isao Sauzedde. Winding and intersection of brownian motions, 2021. arXiv:2112.01645.
+[11] Isao Sauzedde. Lévy area without approximation. Annales de l’Institut Henri Poincaré, Probabilités et Statis-
+tiques, 58(4):2165 – 2200, 2022.
+[12] Wendelin Werner. Rate of explosion of the Amperean area of the planar Brownian loop. In Séminaire de
+Probabilités XXVIII, pages 153–163. Berlin: Springer, 1994.
+[13] Wendelin Werner. Formule de Green, lacet brownien plan et aire de Lévy. Stochastic Process. Appl.,
+57(2):225–245, 1995.
+

ANFKT4oBgHgl3EQfVi5k/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2013819237643118109b793465a9fc261874dd2b761127e18a29fb9f281f0bb2
+size 110695

AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:acdb00fb68525bcedca33e5470e66f44b687263b4d21327837370e5024b38954
+size 1008991

AtAzT4oBgHgl3EQfF_uW/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:11371f8b253074e58a0650d3f7ef217bcc1a9364bd18a3dc6f373ab4bb63cf89
+size 109643

AtFAT4oBgHgl3EQfrx4U/content/2301.08654v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:41ac54d025782cf1b07ee8ee63d9df4313d1012c2edc0cdb0fe0c2881012fb6d
+size 4710469

AtFAT4oBgHgl3EQfrx4U/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:90f6290398470cf927256eb6dc8f31ebcd0c2dde5e1956c32b6a5f7f5d95732f
+size 3997741

AtFAT4oBgHgl3EQfrx4U/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:111ff81d97f949826b3207fe2b7d2ba45133fadffe4e7f6bc71ac6c67f21ff8f
+size 131066

C9E4T4oBgHgl3EQfFwy_/content/tmp_files/2301.04889v1.pdf.txt

@@ -0,0 +1,1665 @@
+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.  
+2 College of Computer Science, Sichuan University, Chengdu 610065, China. 
+3 Tencent AI Lab, Shenzhen 518057, China. 
+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. E-mail: [email protected]. Tel: 86-021-63240090. 
+Xiao Han, Tencent AI Lab, Shenzhen 518057, China. E-mail: [email protected]. Tel: 86-
+075586013388. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+2 
+ 
+ 
+Abstract  
+Background: Clear cell renal cell carcinoma (ccRCC) is the most common renal-related tumor with 
+high heterogeneity. There is still an urgent need for novel diagnostic and prognostic biomarkers for 
+ccRCC. 
+Methods: We proposed a weakly-supervised deep learning strategy using conventional histology 
+of 1752 whole slide images from multiple centers. Our study was demonstrated through internal 
+cross-validation and external validations for the deep learning-based models.  
+Results: Automatic diagnosis for ccRCC through intelligent subtyping of renal cell carcinoma was 
+proved in this study. Our graderisk achieved aera the curve (AUC) of 0.840 (95% confidence interval: 
+0.805-0.871) in the TCGA cohort, 0.840 (0.805-0.871) in the General cohort, and 0.840 (0.805-
+0.871) in the CPTAC cohort for the recognition of high-grade tumor. The OSrisk for the prediction 
+of 5-year survival status achieved AUC of 0.784 (0.746-0.819) in the TCGA cohort, which was 
+further verified in the independent General cohort and the CPTAC cohort, with AUC of 0.774 
+(0.723-0.820) and 0.702 (0.632-0.765), respectively. 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). Kaplan-Meier survival analyses 
+further illustrated that our CRN could significantly distinguish patients with high survival risk, with 
+hazard ratio of 5.664 (3.893-8.239, p < 0.0001) in the TCGA cohort, 35.740 (5.889-216.900, p < 
+0.0001) in the General cohort and 6.107 (1.815 to 20.540, p < 0.0001) in the CPTAC cohort. 
+Comparison analyses conformed that our CRN outperformed current prognosis indicators in the 
+prediction of survival status, with higher concordance index for clinical prognosis. 
+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. 
+ 
+Background  
+Renal-related malignant tumor is one of the most common malignant tumors worldwide. In 2015, 
+the incidence rate of renal cancer arrived at 66.8 per 100,000 in China [1]. In the United States, renal 
+cancer is estimated to have 76,080 new cases and 13,780 associated deaths in 2021 [2]. 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. 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]. However, some reports for 
+ccRCC by experienced pathologists might miss essential elements and lack appropriate information 
+associated prognosis [4]. In addition, traditional diagnosis of ccRCC by pathologist is still time-
+consuming and labor-intensive.  
+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. 
+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. For gigapixel whole-
+slide images (WSIs), they are usually annotated at the slide-level without considering the detailed 
+internal cellular composition. 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]. 
+Some studies have preliminarily demonstrated the utility of weakly-supervised deep learning 
+in kidney segmentation and tumor classification from single center [8, 9]. It is also widely 
+recognized that nuclear grading of cancer cell could act as a prognostic factor for patients with 
+ccRCC [10]. However, traditional assessment with manual observation of nuclear grading may lead 
+to inconsistent judgement between pathologists [11]. Moreover, there are still limitations in current 
+TNM staging system, resulting in an urgent need for novel diagnostic and prognostic biomarkers.  
+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. 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. 
+ 
+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.cancerimagingarchive.net) [12, 13], and the Cancer Genome Atlas (TCGA, 
+https://portal.gdc.cancer.gov) [12] were included. All included patients should meet the following 
+selection criteria: (i) pathologically diagnosed as RCC without other types of malignant tumors; (ii) 
+with corresponding clinical and pathological information (ground-truth label in slide-level); (iii) 
+with access to corresponding H&E slides or their scanned WSIs.  
+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. In addition, 26 patients with renal oncocytoma were also enrolled from Shanghai 
+General Hospital for differential diagnosis analysis. The hematoxylin-eosin staining (H&E)-stained 
+slides were scanned with Leica Aperio AT2 scanners at 20× equivalent magnification. 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. The basic clinical 
+characteristics of the included patients in this study were shown in Supplementary Table S1. Another  
+400 H&E-stained WSIs of RCC from the Pathology AI Platform (PAIP, wisepaip.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).  
+ 
+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. 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. The RCC whole-seg branch aims to learn distinctive features for 
+normal and cancerous regions, which helps to reduce the rate of false positivity. The RCC tumor-seg 
+branch tagetes for the more robust features to recognize tumor regions. 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.  
+
+4 
+ 
+SE-ResNeXt-50 is employed as our encoder, which is a combination of ResNeXt architecture 
+and squeeze-and-excitation (SE) module. 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. For each decoder layer, the trainable transposed convolution operator (TransConv) 
+is used to up-sample feature maps. These features are further connected with features in its 
+corresponding encoder layer via skip-concatenations to preserve the consistently spatial information. 
+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. 
+The output of the decoder is a segmentation map (256 ×256 ×1), indicating the probability of being 
+tumor. The loss function is the combination of segmentation loss (i.e., Dice) and classification loss 
+(binary cross-entropy) in the three branches, which was defined in our previous report [15].  
+ 
+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. 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]. to obtain a descriptive 1024-dimensional feature vector for each patch. These obtained patch-
+level feature vectors are assembled by deep-attention-based pooling to represent the WSI-level 
+feature information. Referring to the attention weight of each patch, the attention pooling would 
+average the representative features of a WSI for prediction. Two fully-connected layers following 
+rectified linear unit (ReLU) are used to conduct WSI-level classification. 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. 
+ 
+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. The overall 
+survival (OS) status at 5-year follow up was used for the training of the prognosis-related models.  
+ 
+Statistical analysis 
+Continuous variants among different groups were analyzed and compared by analysis of variance. 
+The Dice score was set as the evaluation metrics evaluate the performance of our hybrid network in 
+tumor segmentation. Survival analysis was performed via Kaplan–Meier (KM) curve with hazard 
+ratio (HR) and 95% confidence interval (CI) to compare different OS outcomes. We also carried out 
+receiver operating characteristic curve (ROC) analysis with area under curve (AUC) to evaluate the 
+accuracy of the prediction models.  
+ 
+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. Evaluated by five-fold cross-validation in the PAIP cohort, our hybrid 
+network achieved a mean Dice score of 0.796 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. 
+   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. Our hybrid network was generally capable of delineating the boundary 
+between tumor and normal renal tissue with smooth mask (green). 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.e. 
+fiber texture and stroma tissue, were not included in the attentional region (Figure 3B). 
+ 
+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. 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. Based on the strategy, the AUC 
+for distinguishing RCC from normal renal tissue achieved 0.977 (95% CI: 0.969-0.984, Figure 3C) 
+in the in an external validation cohort, which borne comparison with an experienced pathologist.  
+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 0.987 (0.979-0.993), 0.939 (0.913-0.960), and 0.984 (0.961-0.995), respectively 
+(Figure S1), which indicating the robust generalization performance of our model when applied to 
+different scenarios.  
+ 
+Interpretability and whole-slide attention visualization 
+Readable interpretability of deep learning-based clinical models plays important role in further 
+clinical applications [16]. To gain insight into the potential interpretability of our model, we 
+visualized the learned feature space in two dimensions to generate pixel-level heatmaps. As shown 
+in the Figure S1 (right column), the most attended regions recognized by our model were considered 
+to be highly associated with RCC. Areas with red color of the heatmap represented the regions with 
+predicted RCC tissues. 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.  
+ 
+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]. Misconceptions could 
+be reviewed out in clinical practice due to the Review out spectrum of eosinophilic renal neoplasms. 
+Therefore, we further explored whether our hybrid network could be used for the differential 
+diagnosis of RCC from renal oncocytoma in clinical practices. As shown in the Figure S2, our 
+diagnosis model exhibited excellent performance in the differential diagnosis of RCC, which 
+achieved an AUC of 0.951 (0.922-0.972), a sensitivity of 0.821 (0.772-0.862), and a specificity of 
+
+6 
+ 
+0.962 (0.804-0.999).  
+ 
+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]. 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.  
+   As shown in Figure 4A, our subtyping model performed well in the subtype prediction of RCC, 
+with an average AUC of 0.990 (95% CI: 0.981-0.996) in distinguishing ccRCC from pRCC and 
+ChRCC in the TCGA cross-validation cohort, which could be used for the automatic diagnosis of 
+ccRCC. The classification accuracy was further verified in the General cohort, with AUC of 0.970 
+(0.957-0.980, Figure 4B). 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).  
+ 
+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]. Therefore, we further applied the weakly-supervised learning strategy to predict high-grade 
+tumors for the grade-classification of ccRCC. 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. As shown in Figure S3A, our graderisk achieved an average AUC 
+of 0.840 (0.805-0.871) 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 
+0.840 (0.805-0.871, Figure S3C) and 0.840 (0.805-0.871, Figure S3E), respectively. 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.  
+ 
+Intelligent risk quantitation for five-year survival follow-up 
+Clear cell RCC accounts for most of the adverse prognosis related to renal malignancy. 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. 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. The survival risk for 5-year follow-up (OSrisk) was then calculated based on the prediction 
+possibility. As illustrated in Figure S4A, our OSrisk achieved an average AUC of 0.784 (0.746-0.819) 
+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 0.774 (0.723-0.820, Figure S4D) and 0.702 (0.632-0.765, Figure S4G), respectively. 
+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). 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.  
+ 
+
+7 
+ 
+Development of the competing-risk nomogram 
+Integration of multiple biomarkers might improve predictive value over single-scale counterpart [20, 
+21]. 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. 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.  
+We firstly carried out cox regression analysis to identify prognostic indicators. 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. These four factors were further incorporated into the 
+construction of the competing-risk nomogram (CRN, Figure 5B). 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 0.825 (0.789-0.858), specificity of 0.902, and 
+sensitivity of 0.637. With the same cut-off value, patients in the TCGA cohort were classified into 
+the worse group or the favorable group. Kaplan-Meier survival analyses further illustrated that our 
+CRN could significantly distinguish patients with high survival risk (Figure 6A), with HR of 5.664 
+(95% CI 3.893-8.239, p < 0.0001). 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 35.740 (5.889-216.900, p < 0.0001) and 6.107 (1.815 to 20.540, p < 0.0001), respectively.  
+ 
+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). As shown in Table 1, our CRN outperformed 
+current prognosis indicators in the prediction of 5-year, 3-year, 1-year OS status. The CRN achieved 
+the highest C-index value from 0.770 to 0.846, which overmatched current prognosis indicators. 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). 
+ 
+Discussion 
+Traditional visual inspection of pathological images can be distinguished by the nuclear shape, size, 
+nucleolus, and chromatin features. For renal carcinoma with high tumor heterogeneity, traditional 
+microscope vision may miss a lot of important information. Furthermore, the shortage of 
+pathologists has aggravated the presence of overwork in pathology. 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]. There is still an urgent need to develop novel technologies 
+to prevent potential diagnostic error from traditional pathology.  
+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]. 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]. Excellent performance of DL has also been reported in 
+displaying distinct immunogenomic landscape and potential response to immunotherapy [28, 29]. 
+Based on the full landscapes of WSIs, a deep CNN was reported to identify different subtypes 
+
+8 
+ 
+of RCC [30]. A histopathology image classifier could also distinguish TFE3 Xp11.2 translocation 
+RCC from ccRCC, which contributed to overcome the difficulties that could not be easily solved in 
+traditional analysis through naked eye [31]. Benefiting from the increasing number of image 
+datasets, AI-based approaches are now defining integrated and clinically classification of RCC. 
+However, most of the AI-based models were trained from comparatively small samples, without 
+sufficient additional validation. 
+Currently, the diagnosis reports of WSIs are usually at the global level (slide-level). 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.e., inexact 
+supervision). To tackle this problem, we performed the multiple-instance learning to achieve WSI-
+level classification in view of the entire information from the slide. 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. All patches extracted from the same slide were then 
+identified as the instances of a specific WSI. It is noted that WSI were labeled in slide-level 
+annotations of tumor region, and thus, these extracted patches have no annotations. 
+Through the application of CNN, we proposed an end-to-end neural network for the diagnosis 
+and prognosis prediction of RCC. With a WSI input, the network could achieve automatic and rapid 
+diagnosis, grading, and survival prediction for the patient. To our knowledge, this is the largest 
+cohort used in our neural network for the classification of ccRCC using H&E-stained WSIs. 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. In addition, we also provided convincing predictions survival status, 
+which might facilitate clinical decision-making but could not be provided through traditional 
+pathology. 
+In this study, we proposed a data-efficient weakly-supervised learning strategy to address the 
+annotation lack problems in the field of histopathological images. 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]. There are two major differences between this study and ours. First, CLAM adopts 
+pretrained model based on natural images as the feature extractors. The huge domain shift between 
+natural and histopathological images may decrease the model generalization. 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. 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. 
+Several strengths could be found in this study. 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. Secondly, with only a WSI input, the weakly-supervised 
+network makes it possible for automatic and rapid classification for ccRCC. 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. 
+There are also some limitations waiting for solution in our study. Firstly, part of the images 
+
+9 
+ 
+analyzed in this study were acquired for public databases, which might be affected by the potential 
+population bias. Secondly, batch effect might be involved in this analysis since different H&E-
+staining protocols might be performed among different patient cohorts. Thirdly, this is a 
+retrospective study, which might need further validations in prospective clinical studies.  
+ 
+Conclusions 
+In summary, we proposed a weakly-supervised deep learning strategy for the diagnosis and 
+prognosis prediction of RCC with interpretable probability. 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. 
+ 
+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. 
+ 
+Authors’ contributions 
+JHZ and XH conceptualized and supervised the study. STC, XYW, and JZ performed data curation, 
+formal analysis, investigation, visualization, and writing original draft. LRJ, NZ, FG, WY, SY and 
+JXX performed data curation, and validation. All authors involved manuscript editing and 
+manuscript review. 
+ 
+Funding 
+This study was supported by the National Natural Science Foundation of China (81972393). The 
+funders had no role in the design of the study and collection, analysis, and interpretation of data 
+and in writing the manuscript. 
+ 
+Availability of data and materials 
+Primary data are available from Atlas (https://portal.gdc.cancer.gov/) and the Clinical Proteomic 
+Tumor Analysis Consortium (https://www.cancerimagingarchive.net/). Other private data could 
+only be reasonably requested from the corresponding author according to the Research Ethics 
+Committee.  
+ 
+Declarations 
+Ethics approval and consent to participate 
+Our study was approved by the Research Ethics Committee of Shanghai General. Consents were 
+acquired form the participates.  
+ 
+Consent for publication 
+Not applicable. 
+ 
+Competing interests 
+The authors declare that they have no competing interests. 
+ 
+References 
+
+10 
+ 
+[1] Chen W, Zheng R, Baade PD, Zhang S, Zeng H, Bray F, et al. Cancer statistics in China, 2015. 
+CA Cancer J Clin. 2016;66:115-32. 
+[2] Siegel RL, Miller KD, Fuchs HE, Jemal A. Cancer Statistics, 2021. CA Cancer J Clin. 2021;71:7-33. 
+[3] Ljungberg B, Bensalah K, Canfield S, Dabestani S, Hofmann F, Hora M, et al. EAU guidelines on 
+renal cell carcinoma: 2014 update. Eur Urol. 2015;67:913-24. 
+[4] Shuch B, Pantuck AJ, Pouliot F, Finley DS, Said JW, Belldegrun AS, et al. Quality of pathological 
+reporting for renal cell cancer: implications for systemic therapy, prognostication and surveillance. 
+BJU Int. 2011;108:343-8. 
+[5] Chen PC, Gadepalli K, MacDonald R, Liu Y, Kadowaki S, Nagpal K, et al. An augmented reality 
+microscope with real-time artificial intelligence integration for cancer diagnosis. Nat Med. 
+2019;25:1453-7. 
+[6] Ehteshami Bejnordi B, Veta M, Johannes van Diest P, van Ginneken B, Karssemeijer N, Litjens G, 
+et al. Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node 
+Metastases in Women With Breast Cancer. Jama. 2017;318:2199-210. 
+[7] Nagpal K, Foote D, Liu Y, Chen PC, Wulczyn E, Tan F, et al. Development and validation of a 
+deep learning algorithm for improving Gleason scoring of prostate cancer. NPJ Digit Med. 
+2019;2:48. 
+[8] Han S, Hwang SI, Lee HJ. The Classification of Renal Cancer in 3-Phase CT Images Using a Deep 
+Learning Method. J Digit Imaging. 2019;32:638-43. 
+[9] Tanaka T, Huang Y, Marukawa Y, Tsuboi Y, Masaoka Y, Kojima K, et al. Differentiation of Small 
+(≤ 4 cm) Renal Masses on Multiphase Contrast-Enhanced CT by Deep Learning. AJR Am J 
+Roentgenol. 2020;214:605-12. 
+[10] Delahunt B, Eble JN, Egevad L, Samaratunga H. Grading of renal cell carcinoma. 
+Histopathology. 2019;74:4-17. 
+[11] Holdbrook DA, Singh M, Choudhury Y, Kalaw EM, Koh V, Tan HS, et al. Automated Renal 
+Cancer Grading Using Nuclear Pleomorphic Patterns. JCO Clin Cancer Inform. 2018;2:1-12. 
+[12] Clark K, Vendt B, Smith K, Freymann J, Kirby J, Koppel P, et al. The Cancer Imaging Archive 
+(TCIA): maintaining and operating a public information repository. J Digit Imaging. 2013;26:1045-
+57. 
+[13] Clark DJ, Dhanasekaran SM, Petralia F, Pan J, Song X, Hu Y, et al. Integrated Proteogenomic 
+Characterization of Clear Cell Renal Cell Carcinoma. Cell. 2019;179:964-83.e31. 
+[14] Li X, Wang W, Hu X, Yang J. Selective Kernel Networks.  2019 IEEE/CVF Conference on 
+Computer Vision and Pattern Recognition (CVPR)2020. 
+[15] Wang X, Fang Y, Yang S, Zhu D, Wang M, Zhang J, et al. A hybrid network for automatic 
+hepatocellular carcinoma segmentation in H&E-stained whole slide images. Med Image Anal. 
+2021;68:101914. 
+[16] Lu MY, Williamson DFK, Chen TY, Chen RJ, Barbieri M, Mahmood F. Data-efficient and weakly 
+supervised computational pathology on whole-slide images. Nat Biomed Eng. 2021;5:555-70. 
+[17] Amin MB, Crotty TB, Tickoo SK, Farrow GM. Renal oncocytoma: a reappraisal of morphologic 
+features with clinicopathologic findings in 80 cases. Am J Surg Pathol. 1997;21:1-12. 
+[18] Patard JJ, Leray E, Rioux-Leclercq N, Cindolo L, Ficarra V, Zisman A, et al. Prognostic value of 
+histologic subtypes in renal cell carcinoma: a multicenter experience. J Clin Oncol. 2005;23:2763-
+71. 
+[19] Delahunt B, Cheville JC, Martignoni G, Humphrey PA, Magi-Galluzzi C, McKenney J, et al. The 
+
+11 
+ 
+International Society of Urological Pathology (ISUP) grading system for renal cell carcinoma and 
+other prognostic parameters. Am J Surg Pathol. 2013;37:1490-504. 
+[20] Chen D, Liu Z, Liu W, Fu M, Jiang W, Xu S, et al. Predicting postoperative peritoneal metastasis 
+in gastric cancer with serosal invasion using a collagen nomogram. Nat Commun. 2021;12:179. 
+[21] Jiang Y, Zhang Q, Hu Y, Li T, Yu J, Zhao L, et al. ImmunoScore Signature: A Prognostic and 
+Predictive Tool in Gastric Cancer. Ann Surg. 2018;267:504-13. 
+[22] Metter DM, Colgan TJ, Leung ST, Timmons CF, Park JY. Trends in the US and Canadian 
+Pathologist Workforces From 2007 to 2017. JAMA Netw Open. 2019;2:e194337. 
+[23] Barisoni L, Lafata KJ, Hewitt SM, Madabhushi A, Balis UGJ. Digital pathology and 
+computational image analysis in nephropathology. Nat Rev Nephrol. 2020;16:669-85. 
+[24] Pantanowitz L, Quiroga-Garza GM, Bien L, Heled R, Laifenfeld D, Linhart C, et al. An artificial 
+intelligence algorithm for prostate cancer diagnosis in whole slide images of core needle biopsies: 
+a blinded clinical validation and deployment study. Lancet Digit Health. 2020;2:e407-e16. 
+[25] Lu MY, Chen TY, Williamson DFK, Zhao M, Shady M, Lipkova J, et al. AI-based pathology 
+predicts origins for cancers of unknown primary. Nature. 2021;594:106-10. 
+[26] Courtiol P, Maussion C, Moarii M, Pronier E, Pilcer S, Sefta M, et al. Deep learning-based 
+classification of mesothelioma improves prediction of patient outcome. Nat Med. 2019;25:1519-
+25. 
+[27] Coudray N, Ocampo PS, Sakellaropoulos T, Narula N, Snuderl M, Fenyö D, et al. Classification 
+and mutation prediction from non-small cell lung cancer histopathology images using deep 
+learning. Nat Med. 2018;24:1559-67. 
+[28] Xie F, Zhang J, Wang J, Reuben A, Xu W, Yi X, et al. Multifactorial Deep Learning Reveals Pan-
+Cancer Genomic Tumor Clusters with Distinct Immunogenomic Landscape and Response to 
+Immunotherapy. Clin Cancer Res. 2020;26:2908-20. 
+[29] Sealfon RSG, Mariani LH, Kretzler M, Troyanskaya OG. Machine learning, the kidney, and 
+genotype-phenotype analysis. Kidney Int. 2020;97:1141-9. 
+[30] Marostica E, Barber R, Denize T, Kohane IS, Signoretti S, Golden JA, et al. Development of a 
+Histopathology Informatics Pipeline for Classification and Prediction of Clinical Outcomes in 
+Subtypes of Renal Cell Carcinoma. Clin Cancer Res. 2021;27:2868-78. 
+[31] Cheng J, Han Z, Mehra R, Shao W, Cheng M, Feng Q, et al. Computational analysis of 
+pathological images enables a better diagnosis of TFE3 Xp11.2 translocation renal cell carcinoma. 
+Nat Commun. 2020;11:1778. 
+[32] Wang X, Du Y, Yang S, et al. RetCCL: Clustering-guided contrastive learning for whole-slide 
+image retrieval[J]. Medical Image Analysis, 2023, 83: 102645.. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+12 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Table 1 Comparison with current prognosis indicators 
+ 
+5-year OS status 
+3-year OS status 
+1-year OS status 
+C-index 
+ 
+AUC 
+95% CI 
+AUC 
+95% CI 
+AUC 
+95% CI 
+ 
+TCGA cohort 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Grade 
+0.708 
+0.666-0.747 
+0.718 
+0.676-0.757 
+0.726 
+0.685-0.764 
+0.667 
+Stage 
+0.764 
+0.724-0.800 
+0.794 
+0.756-0.828 
+0.822 
+0.786-0.854 
+0.729 
+Grade risk 
+0.723 
+0.682-0.762 
+0.733 
+0.692-0.771 
+0.725 
+0.684-0.763 
+0.677 
+OS risk 
+0.785 
+0.747-0.820 
+0.779 
+0.740-0.814 
+0.812 
+0.775-0.845 
+0.727 
+CRN 
+0.825 
+0.789-0.858 
+0.841 
+0.806-0.871 
+0.869 
+0.837-0.897 
+0.770 
+General cohort 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Grade 
+0.798 
+0.748-0.841 
+0.848 
+0.802-0.886 
+0.943 
+0.910-0.966 
+0.820 
+Stage 
+0.788 
+0.738-0.833 
+0.842 
+0.796-0.881 
+0.856 
+0.812-0.893 
+0.800 
+Grade risk 
+0.799 
+0.750-0.843 
+0.880 
+0.839-0.915 
+0.882 
+0.841-0.916 
+0.820 
+OS risk 
+0.774 
+0.723-0.820 
+0.885 
+0.844-0.919 
+0.870 
+0.828-0.906 
+0.803 
+CRN 
+0.814 
+0.766-0.856 
+0.924 
+0.888-0.951 
+0.969 
+0.943-0.986 
+0.846 
+CPTAC cohort 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Grade 
+0.677 
+0.607-0.742 
+0.694 
+0.624-0.758 
+0.627 
+0.556-0.695 
+0.659 
+Stage 
+0.796 
+0.733-0.850 
+0.803 
+0.740-0.856 
+0.748 
+0.680-0.807 
+0.773 
+Grade risk 
+0.693 
+0.624-0.757 
+0.711 
+0.642-0.773 
+0.685 
+0.615-0.750 
+0.689 
+OS risk 
+0.702 
+0.632-0.765 
+0.708 
+0.639-0.771 
+0.679 
+0.609-0.744 
+0.684 
+CRN 
+0.803 
+0.740-0.856 
+0.809 
+0.747-0.862 
+0.754 
+0.688-0.813 
+0.780 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+13 
+ 
+ 
+ 
+ 
+ 
+Figures 
+ 
+ 
+Figure 1 The workflow and the architecture of the hybrid neural network proposed in this study. Br 
+whole _ seg, pixel-wise renal cell carcinoma segmentation; Br cls, auxiliary classification task.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Br
+tumor seg
+Br
+Br
+cls
+whole seg14 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2 Architecture of weakly-supervised learning strategy based on multiple instance-learning 
+scene for subtype diagnosis, grade staging, and survival analysis of renal cell carcinoma. FC, fully 
+connected.  
+ 
+ 
+
+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<0.000115 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3 Accurate segmentation of RCC for intelligent diagnosis. (A) Example RCC segmentation. 
+Left, original slide image; Right, recognized slide image with green curve. (B) Example of 
+segmentation on different kinds of tissue. (C) ROC curve for distinguishing RCC from normal renal 
+tissues in the independent verification cohort. RCC, renal cell carcinoma; ROC, receiver operator 
+characteristics; AUC, area under the curve (with 95% confidence interval).  
+ 
+ 
+ 
+
+A
+C
+Sensitivity (%)
+4
+2
+AUC:0.977(0.969-0.984)
+Sensitivity:0.973(0.963-0.981)
+Specificity:0.902(0.846-0.943)
+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. (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. (C) Visualizations of the 
+diagnosis for ccRCC. The detected tumor regions were shown in red. 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; CI, confidence interval.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+A
+B
+100
+100
+8
+8
+Sensitivity (%)
+Sensitivity (%)
+09
+4
+4
+AUC
+95%CI
+2
+AUC
+95%CI
+2
+CcRCC:0.990(0.981-0.996)
+CcRCC:0.970(0.957-0.980）
+pRCC:0.995 (0.987-0.999)
+pRCC: 0.995 (0.978-0.997)
+0
+ChRCC:0.992(0.980-0.998)
+0
+ChRCC:0.935（0.897-0.999）
+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. (A) Cox regression analyses of the deep 
+learning-based pathology signature and clinicopathological features. (B) The competing-risk 
+nomogram for the construction of the prognosis prediction model combining the deep learning-
+based pathology signature and clinicopathological features. TCGA, the Cancer Genome Atlas; 
+CPTAC, Clinical Proteomic Tumor Analysis Consortium; OS, overall survival.  
+ 
+ 
+ 
+ 
+ 
+
+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<0.05
+●p<0.05
+●p<0.05
+OS risk
+o p>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 
+ 
+ 
+ 
+ 
+ 
+

CdA0T4oBgHgl3EQfAf80/content/2301.01962v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1f8d206809096abfdc6731caf2657f780b196aac782e11aac25db17388d4ae7e
+size 8448073

D9E1T4oBgHgl3EQfWgQs/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:36d974e8ba6e8155d6bf5918c00039bdb2bf9d474a7afb7287da64dd88b3fc2b
+size 38404141

EdAyT4oBgHgl3EQfSfdI/content/tmp_files/2301.00086v1.pdf.txt

@@ -0,0 +1,823 @@
+Phase-Slip Lines and Anomalous Josephson Effects in a
+Tungsten Clusters-Topological Insulator Microbridge
+Dong-Xia Qu1, Joseph J. 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. Phys. 82, 3045
+(2010).
+[4] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057
+(2011).
+[5] M.-X. Wang, et al., The coexistence of superconductivity and topological order in the Bi2Se3 thin
+films, Science 336, 52 (2012).
+[6] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E.
+M. Hankiewicz, L. W. Molenkamp, B. I. Halperin, and A. Yacoby, Topological superconductivity
+in a phase-controlled Josephson junction, Nature 569, 93 (2019).
+[7] M. Veldhorst et al., Josephson supercurrent through a topological insulator surface state, Nat.
+Mat. 11, 417 (2012).
+[8] F. Qu et al., Strong superconducting proximity effect in PbBi2Te3 hybridstructures, Sci. Rep. 2,
+339 (2012).
+[9] J. R. Williams, A. J. Bestwick, P. Gallagher, S. S. Hong, Y. Cui, A. S. Bleich, J. G. Analytis, I.
+R. Fisher, and D. Goldhaber-Gordon, Unconventional Josephson effect in hybrid superconductor-
+topological insulator devices, Phys. Rev. Lett. 109, 056803 (2012).
+[10] C. G. Molenaar, D. P. Leusink, X. L. Wang, and A. Brinkman, Geometric dependence of Nb-
+Bi2Te3-Nb topological Josephson junction transport parameters, Supercond. Sci. Technol. 27,
+104003, (2014).
+[11] S. Charpentier et al, Induced unconventional superconductivity on the surface states of Bi2Te3
+topological insulator, Nat. Commun. 8, 2019 (2017).
+[12] J. Wiedenmann et al., 4π-periodic Josephson supercurrent in HgTe-based topological Josephson
+junctions, Nat. Commun. 7, 10303 (2016).
+[13] L. Galletti et al., Influence of topological edge states on the properties of Al/Bi2Se3/Al hybrid
+Josephson devices, Phys. Rev. B 89, 134512 (2014).
+[14] P. Sch¨uffelgen et al, Selective area growth and stencil lithography for in situ fabricated quantum
+devices, Nat. Nano. 14, 825 (2019).
+[15] D. Rosenbach et al, Reappearance of first Shapiro step in a narrow topological Josephson junctions,
+Sci. Adv. 7, eabf1854 (2021).
+[16] M. Bai et al, Proximity-induced superconductivity in (Bi1−xSbx)2Te3 topological-insulator
+nanowires, Commun. Mater. 3, 20 (2022).
+[17] I. T. Rosen et al, Fractional AC Josephson effect in a topological insulator proximitized by a
+self-formed superconductor, arXiv:2110.01039 (2021).
+[18] J. J. Cuozzo et al., Leggett Modes in Dirac Semimetals, arXiv:2205.15995 (2022).
+[19] B. H. Elfeky et al, Reemergence of missing Shapiro steps in the presence of in-plane magnetic
+field, arXiv:2210.06502 (2022).
+[20] M. C. Dartiailh, J. J. Cuozzo, B. H. Elfeky, W. Mayer, J. Yuan, K. S. Wickramaasinghe, E.
+Rossi, and J. Shabani, Missing Shapiro steps in topologically trivial Josephson junction on InAs
+quantum well, Nat. Commun. 12, 78 (2021).
+11
+
+[21] D.-X. Qu, S. K. Roberts, and G. F. Chapline, Observation of huge surface hole mobility in the
+topological insulator Bi0.91Sb0.09 (111), Phys. Rev. Lett. 111, 176801 (2013).
+[22] D.-X. Qu, N. E. Teslich, Z. Dai, G. F. Chapline, T. Schenkel, S. R. Durham, and J. Dubois, Onset
+of two-dimensional superconducting phase in a topological-insulator-normal-metal Bi1−xSbx/Pt
+junction fabricated by ion-beam techniques, Phys. Rev. Lett., 121, 037001 (2018).
+[23] R. Vaglio, C. Attanasio, L. Maritato, and A. Ruosi, Explanation of the resistance-peak anomaly
+in nonhomogeneous superconductors, Phys. Rev. B 47, 15302 (1993).
+[24] S. Eley, S. Gopalakrishnan, P. M. Goldbart, and N. Mason, Approaching zero-temperature metallic
+states in mesoscopic superconductor-normal-superconductor arrays, Nat. Phys. 8, 59 (2012).
+[25] D. J. Resnick, J. C. Garland, J. T. Boyd, S. Shoemaker, and R. S. Newrock, Kosterlitz-Thouless
+Transition in Proximity-Coupled Superconducting Arrays, Phys. Rev. Lett. 47, 1542 (1981).
+[26] J. M. Kosterlitz and D. J. Thouless, Ordering, metastability, and phase transitions in two-
+dimensional systems, J. Phys. C Solid State Phys. 6, 1181 (1973).
+[27] B. I. Halperin and D. R. Nelson, Resistive transition in superconducting films, J. Low Temp.
+Phys. 36, 599 (1979).
+[28] G. Chapline, Superfluid transition in a chiron gas, Philos. Mag. 88, 1227, (2008).
+[29] Y. Sun, H. Xiao, M. Zhang, Z. Xue, Y. Mei, X. Xie, T. Hu, Z. Di, and X. Wang, Double quantum
+criticality in superconducting tin arrays-graphene hybrid, Nat. Commun. 9, 2159 (2018).
+[30] A. G. Sivakov, A. M. Glukhov, and A. N. Omelyanchouk, Y. Koval, P. Muller, and A.V. Ustinov,
+Josephson behavior of phase-slip lines in wide superconducting strips, Phys. Rev. Lett. 91, 267001
+(2003).
+[31] V. M. Dmitriev, I. V. Zolochevskii, T. V. Salenkova, and E. V. Khristenko, Critical currents, phase
+slip centers, and phase slip lines in superconducting films in the absence of external magnetic field,
+Low Temp. Phys. 31, 127 (2005).
+[32] W. J. Skocpol, M. R. Beasley, and M. Tinkham, Phase-slip centers and nonequilibrium processes
+in superconducting tin microbridges. J. Low Temp. Phys. 16, 145 (1974).
+[33] V. G. Volotskaya, I. M. Dmitrenko,L. E. Musienko, and A. G. Sivakov, Current-induced destruc-
+tion of superconductivity in wide films Sov. J. Low Temp. Phys. 7, 188 (1981).
+[34] I. M. Dmitrenko, Resistive state of broad superconducting films and phase-slip lines (a review),
+Low Temp. Phys. 22, 648 (1996).
+[35] A. Andronov, I. Gordion, V. Kurin, I. Nefedov and I. Shereshevsky, Kinematic vortices and phase
+slip lines in the dynamics of the resistive state of narrow superconductive thin film channels,
+Physica C 213, 193 (1993).
+[36] G. Berdiyorov, K. Harrabi, F. Oktasendra, K. Gasmi, A. I. Mansour, J. P. Maneval, and F. M.
+Peeters, Dynamics of current-driven phase-slip centers in superconducting strips, Phys. Rev. B
+90, 054506 (2014).
+[37] S. Shapiro, Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and
+Other Observations, Phys. Rev. Lett. 11, 80 (1963).
+[38] H.-J. Kwon, K. Sengupta, V. M. Yakovenko, Fractional ac Josephson effect in p- and d-wave
+superconductors, Eur. Phys. J. B 37, 349 (2004).
+[39] A. Y. Kitaev, Unpaired Majorana fermions in quantum wires. Phys.-Uspekhi 44, 131 (2001).
+[40] L. P. Rokhinson, X. Liu, and J. K. Furdyna, The fractional a.c. Josephson effect in a
+semiconductor-superconductor nanowire as a signature of Majorana particles, Nat. Phys. 8, 795
+(2012).
+12
+
+[41] E. Bocquillon, R. S. Deacon, J. Wiedenmann, P. Leubner, T. M. Klapwijk, C. Br¨une, K. Ishibashi,
+H. Buhmann, and L. W. Molenkamp, Gapless Andreev bound states in the quantum spin Hall
+insulator HgTe, Nat. Nano. 12, 137 (2017).
+[42] R. Deacon et al., Josephson Radiation from Gapless Andreev Bound States in HgTe-Based Topo-
+logical Junctions, Phys. Rev. X 7, 021011 (2017).
+[43] D. Laroche et al., Observation of the 4π-periodic Josephson effect in indium arsenide nanowires,
+Nat. Commu. 10, 245 (2019).
+[44] F. Dom´ınguez, O. Kashuba, E. Bocquillon, J. Wiedenmann, R. S. Deacon, T. M. Klapwijk, G.
+Platero, L. W. Molenkamp, B. Trauzettel, and E. M. Hankiewiczl, Josephson junction dynamics
+in the presence of 2π- and 4π-periodic supercurrents, Phys. Rev. B 95, 195430 (2017).
+[45] B. Rossignol, T. Kloss, and W. Waintal, Role of Quasiparticles in an Electric Circuit with Joseph-
+son Junctions, Phys. Rev. Lett. 122, (2019).
+[46] F. Dom´ınguez, F. Hassler, and G. Platero, Dynamical detection of Majorana fermions in current-
+biased nanowires, Phys. Rev. B 86, 140503(R) (2012).
+[47] D. Averin and A. Bardas, ac Josephson effect in a single quantum channel, Phys. Rev. Lett. 75,
+1831 (1995).
+[48] N. Paradiso, A.-T. Nguyen, K. E. Kloss, and C. Strunk, Phase slip lines in superconducting
+few-layer NbSe2 crystals, 2D Materials 6, 025039 (2019).
+[49] F. Nichele et al., Scaling of Majorana Zero-Bias Conductance Peaks, Phys. Rev. Lett. 119, 136803
+(2017).
+[50] C. Huang et al., Proximity-induced surface superconductivity in Dirac semimetal Cd3As2, Nat.
+Commu. 10, 2217 (2019).
+13
+

FdE4T4oBgHgl3EQffw1T/content/2301.05110v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2bf56303e239c5818da3940ec2df33065fd060bae5b6c8995ae2292b8fed6c5e
+size 1304102

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: [email protected] (Werner M. Seiler),
+[email protected] (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. Furthermore, the simple shooting method works very
+well and reliable for this class of problems.
+Acknowledgements
+This work was performed within the Research Training Group Biological
+Clocks on Multiple Time Scales (GRK 2749) at Kassel University funded by the
+German Research Foundation (DFG).
+26
+
+References
+[1] R. Emden, Gaskugeln, Teubner, Leipzig, 1907.
+[2] H. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York,
+1962.
+[3] C. Hansen, S. Kawaler, V. Trimble, Stellar Interiors, 2nd Edition, Astronomy and Astro-
+physics Library, Springer-Verlag, New York, 2004.
+[4] G. Horedt, Polytropes — Applications in Astrophysics and Related Fields, Astrophysics
+and Space Science Library 306, Kluwer, New York, 2004.
+[5] L. Thomas, The calculation of atomic fields, Proc. Cambr. Philos. Soc. 23 (1927) 542–548.
+[6] E. Fermi, Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms
+und ihre Anwendung auf die Theorie des periodischen Systems der Elemente, Z. Phys.
+48 (1928) 73–79.
+[7] E. Di Grezia, S. Esposito, Fermi, Majorana and the statistical model of atoms, Found. Phys.
+34 (2004) 1431–1450.
+[8] N. March, The Thomas–Fermi approximation in quantum mechanics, Adv. Phys. 6 (1957)
+1–101.
+[9] I. Torrens, Interatomic Potentials, Academic Press, New York, 1972.
+[10] E. Hille, On the Thomas–Fermi equation, Proc. Natl. Acad. Sci. USA 62 (1969) 7–10.
+[11] E. Hille, Some aspects of the Thomas–Fermi equation, J. Anal. Math. 23 (1970) 147–170.
+[12] G. Horedt, Seven-digit tables of Lane–Emden functions, Astrophys. Space Sci. 126 (1986)
+357–408.
+[13] R. Russell, L. Shampine, Numerical methods for singular boundary value problems, SIAM
+J. Num. Anal. 12 (1975) 13–36.
+[14] V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd Edi-
+tion, Grundlehren der mathematischen Wissenschaften 250, Springer-Verlag, New York,
+1988.
+[15] A. Remizov, Multidimensional Poincar´e construction and singularities of lifted fields for
+implicit differential equations, J. Math. Sci. 151 (2008) 3561–3602.
+[16] W. Seiler, Singularities of implicit differential equations and static bifurcations, in:
+V. Gerdt, W. Koepf, E. Mayr, E. Vorozhtsov (Eds.), Computer Algebra in Scientific Com-
+puting — CASC 2013, Lecture Notes in Computer Science 8136, Springer-Verlag, Cham,
+2013, pp. 355–368.
+[17] W. Seiler, M. Seiß, Singular initial value problems for scalar quasi-linear ordinary differ-
+ential equations, J. Diff. Eq. 281 (2021) 258–288.
+[18] E. Braun, W. Seiler, M. Seiß, On the numerical analysis and visualisation of implicit ordi-
+nary differential equations, Math. Comput. Sci. 14 (2020) 281–293.
+[19] W. Seiler, Involution — The Formal Theory of Differential Equations and its Applications
+in Computer Algebra, Algorithms and Computation in Mathematics 24, Springer-Verlag,
+Berlin, 2010.
+[20] U. Kant, W. Seiler, Singularities in the geometric theory of differential equations, in:
+W. Feng, Z. Feng, M. Grasselli, X. Lu, S. Siegmund, J. Voigt (Eds.), Dynamical Systems,
+Differential Equations and Applications (Proc. 8th AIMS Conference, Dresden 2010),
+Vol. 2, AIMS, 2012, pp. 784–793.
+[21] M. Lange-Hegermann, D. Robertz, W. Seiler, M. Seiß, Singularities of algebraic differential
+equations, Adv. Appl. Math. 131 (2021) 102266.
+27
+
+[22] W. Beyn, W. Kleß, Numerical Taylor expansion of invariant manifolds in large dynamical
+systems, Numer. Math. 80 (1998) 1–38.
+[23] T. Eirola, J. von Pfaler, Taylor expansion for invariant manifolds, Numer. Math. 80 (1998)
+1–38.
+[24] J. Liang, A singular initial value problem and self-similar solutions of a nonlinear dissipa-
+tive wave equation, J. Diff. Eqs. 246 (2009) 819–844.
+[25] J. Sijbrand, Properties of center manifolds, Trans. AMS 289 (1985) 431–469.
+[26] W. Seiler, M. Seiß, Singular initial value problems for quasi-linear systems of ordinary
+differential equations, in preparation (2022).
+[27] G. Vainikko, A smooth solution to a linear system of singular ODEs, Zeitsch. Analysis
+Anwend. 32 (2013) 349–370.
+[28] G. Vainikko, A smooth solution to a nonlinear system of singular ODEs, in: T. Simos,
+G. Psihoyios, C. Tsitouras (Eds.), Proc. 11th Int. Conf. Numerical Analysis and Applied
+Mathematics (ICNAAM 2013), AIP Conf. Proc. 1558, Amer. Inst. Physics, 2013, pp. 758–
+761.
+[29] M. Chawla, P. Shivakumar, On the existence of solutions of a class of singular nonlinear
+two-point boundary value problems, J. Comp. Appl. Math. 19 (1987) 379–388.
+[30] K. Parand, M. Deghan, A. Rezaei, S. Ghaderi, An approximation algorithm for the solu-
+tion of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite
+functions collocation method, Comp. Phys. Comm. 181 (2010) 1096–1108.
+[31] R. Aris, Introduction to the Analysis of Chemical Reactors, International Series in the
+Physical and Chemical Engineering Sciences, Prentice-Hall, Englewood Cliffs, 1965.
+[32] D. Flockerzi, K. Sundmacher, On coupled Lane–Emden equations arising in dusty fluid
+models, J. Phys. Conf. Ser. 268 (2011) 012006.
+[33] T. Praveen, P. Valencia, L. Rajendran, Theoretical analysis of intrinsic reaction kinetics and
+the behavior of immobilized enzymes system for steady-state conditions, Biochem. Eng. J.
+91 (2014) 129–139.
+[34] A. Ananthaswamy, R. Shanthakumari, M. Subha, Simple analytical expressions of the non-
+linear reaction diffusion process in an immobilized biocatalyst particle using the new ho-
+motopy perturbation method, Rev. Bioinform. Biometr. 3 (2014) 22–28.
+[35] L. Shampine, J. Kierzenka, A BVP solver based on residual control and the MATLAB PSE,
+ACM Trans. Math. Softw. 27 (2001) 299–316.
+[36] L. Shampine, J. Kierzenka, A BVP solver that controls residual and error, J. Numer. Anal.
+Ind. Appl. Math. 3 (2008) 27–41.
+[37] R. Skeel, M. Berzins, A method for the spatial discretization of parabolic equations in one
+space variable, SIAM J. Sci. Stat. Comp. 11 (1990) 1–32.
+[38] S. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor.
+Biol. 60 (1976) 449–457.
+[39] D. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis–
+Menten oxygen uptake kinetics, J. Theor. Biol. 71 (1978) 255–263.
+[40] P. Hiltmann, P. Lory, On oxygen diffusion in spherical cell with Michaelis–Menten uptake
+kinetics, Bull. Math. Biol. 45 (1983) 661–664.
+[41] S. Khury, A. Sayfy, A novel approach for the solution of a class of singular boundary value
+problems arising in physiology, Math. Comp. Modell. 52 (2010) 626–636.
+[42] H. C¸ a˘glar, N. C¸ a˘glar, M. ¨Ozer, B-spline solution of non-linear singular boundary value
+problems arising in physiology, Chaos Solit. Fract. 39 (2009) 1232–1237.
+28
+
+[43] F. de Hoog, R. Weiss, Difference methods for boundary value problems with a singularity
+of the first kind, SIAM J. Numer. Anal. 13 (1976) 775–813.
+[44] M. Campesi, N. Mariani, S. Bressa, M. Pramparo, B. Barbero, L. Cad´us, G. Baretto,
+O. Mart´ınez, Kinetic study of the combustion of ethanol and ethyl acetate mixtures over
+a MnCu catalyst, Fuel Proc. Technol. 103 (2012) 84–90.
+[45] V. Meena, T. Praveen, L. Rajendran, Mathematical modeling and analysis of the molar
+concentrations of ethanol, acetaldehyde and ethyl acetate inside the catalyst particle, Kinet.
+Catal. 57 (2016) 125–134.
+[46] S. Bressa, N. Mariani, N. Ardiaca, G. Mazza, O. Mart´ınez, G. Barreto, An algorithm for
+evaluating reaction rates of catalytic reaction networks with strong diffusion limitations,
+Comp. Chem. Eng. 25 (2001) 1185–1198.
+[47] S. Esposito, Majorana solution of the Thomas–Fermi equation, Amer. J. Phys. 70 (2002)
+852���856.
+[48] S. Esposito, Majorana transformation for differential equations, Int. J. Theor. Phys. 41
+(2002) 2417–2426.
+[49] A. Mambriani, Su un teorema relativo alle equazioni differenziali ordinarie del 2◦ ordine,
+Rend. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (Ser. VI) 9 (1929) 620–629.
+[50] K. Parand, M. Delkosh, Accurate solution of the Thomas–Fermi equation using the frac-
+tional order of rational Chebyshev functions, J. Comp. Appl. Math. 317 (2017) 624–642.
+[51] F. Fern´andez, J. Garcia, On the Majorana solution to the Thomas–Fermi equation,
+arXiv:2105.02686 (2021).
+[52] P. Amore, J. Boyd, F. Fern´andez, Accurate calculation of the solutions to the Thomas–
+Fermi equations, Appl. Math. 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
+

G9E4T4oBgHgl3EQfHwwY/content/2301.04905v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f308603ef9398b0e620063aa4abd866dd6c623a17ac34bed9d49749ff9f5c213
+size 246699

G9E4T4oBgHgl3EQfHwwY/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2b9c88bc66217fcd1a6b48aba2b47e5f8b5d258dcefedfae394372005d00baa9
+size 2424877

G9E4T4oBgHgl3EQfHwwY/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a86041f4eecf02e021141adaa26fd46111da6c962f16b3888888bd0bff5da6e5
+size 104108

HNA0T4oBgHgl3EQfBv_D/content/tmp_files/2301.01981v1.pdf.txt

@@ -0,0 +1,156 @@
+ 
+1 
+ 
+ 
+Comment on “Biological modeling of gold 
+nanoparticle enhanced radiotherapy for proton 
+therapy” by Lin et al. [Phys. Med. Biol. 60 (2015) 
+4149–4168] 
+Hans Rabus 1 
+1 Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany 
+ 
+E-mail: [email protected] 
+ 
+ 
+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 
+ 
+

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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' MD: National Institute of Standards and Technology) Jones B L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' McMahon S J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Li W B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Villagrasa C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Schuemann J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Hepperle P A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' de la Fuente Rosales L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Beuve M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Maria S D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Klapproth A P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Li C Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' Poignant F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}
+page_content=' 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'}

INAyT4oBgHgl3EQfffiq/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:55aa945e535b7c9d6e578e2bb0184328c87b3d1bd53610ef54fb32472e2dafc5
+size 168014